SPERNER THEOREMS FOR UNRELATED COPIES OF POSETS AND GENERATING DISTRIBUTIVE LATTICES Текст научной статьи по специальности «Философия, этика, религиоведение»

URAL MATHEMATICAL JOURNAL, Vol. 10, No. 1, 2024, pp. 44-60

DOI: 10.15826/umj.2024.1.004

SPERNER THEOREMS FOR UNRELATED COPIES OF POSETS AND GENERATING DISTRIBUTIVE LATTICES12

Gabor Czedli

Bolyai Institute, University of Szeged, Aradi vertanuk tere 1, H-6720 Szeged, Hungary

czedli@math.u-szeged.hu

Abstract: For a finite poset (partially ordered set) U and a natural number n, let S(U, n) denote the largest number of pairwise unrelated copies of U in the powerset lattice (AKA subset lattice) of an n-element set. If U is the singleton poset, then S(U, n) was determined by E. Sperner in 1928; this result is well known in extremal combinatorics. Later, exactly or asymptotically, Sperner's theorem was extended to other posets by A. P. Dove, J. R. Griggs, G. O. H. Katona, D. J. Stahl, and W. T. Jr. Trotter. We determine S(U, n) for all finite posets with 0 and 1, and we give reasonable estimates for the "V-shaped" 3-element poset and, mainly, for the 4-element poset with 0 and three maximal elements. For a lattice L, let Gmin(L) denote the minimum size of generating sets of L. We prove that if U is the poset of the join-irreducible elements of a finite distributive lattice D, then the function k ^ Gm;n(Dk) is the left adjoint of the function n ^ S(U, n). This allows us to determine Gmin(Dk) in many cases. E.g., for a 5-element distributive lattice D, Gmin(D2023) = 18 if D is a chain and Gmin(D2023) = 15 otherwise. The present paper, another recent paper, and a 2021 one indicate that large direct powers of small distributive lattices could be of interest in cryptography.

Keywords: Sperner theorem for partially ordered sets, Antichain of posets, Unrelated copies of a poset, Incomparable copies of a poset, Distributive lattice, Smallest generating set, Minimum-sized Generating set, Cryptography with lattices.

1. Introduction

This paper belongs both to extremal combinatorics and lattice theory, and it is intended to be self-contained for those who know the concept of a free semilattice, that of a distributive lattice, and the relation between lattice orders and lattice operations.

Our main goal is to establish a bridge between the combinatorial topic of Sperner (type) theorems and the lattice theoretical topic of minimum generating sets of finite lattices; this goal is accomplished by Theorem 1 in Section 2. If we start from the Sperner (type) theorems proved by Griggs, Stahl, and Trotter [9], Dove and Griggs [6], and Katona and Nagy [10], then the just-mentioned "bridge" can lead only to asymptotic results, in which we are less interested, or to rather special distributive lattices. Hence, we modestly generalize their Sperner theorems, see Observation 1, and we give reasonable estimates for a particular case; see Proposition 1.

A poset (that is, partially ordered set) U is said to be bounded if it has a smallest element, denoted by 0 = 0U, and a largest element, 1 = 1U; these elements are uniquely determined if they both exist. In Section 3, we give an exact formula for the maximum number of pairwise unrelated isomorphic copies of a finite bounded poset among the subsets of an n-element set; see Observation 1, which is an easy generalization of a result of Griggs, Stahl, and Trotter [9] from chains to bounded posets. The situation becomes more exciting in Section 4, where we present estimates for two particular posets, V and W given in Fig. 1.

1This research was supported by the National Research, Development and Innovation Fund of Hungary, under funding scheme K 138892.

2 This paper is dedicated, to my colleague Eszter K. Horvath, PhD, on her sixtieth birthday.

ô

V

D=Dn( V)

W

Dn( W)

Figure 1. Two posets and the corresponding distributive lattices

The search for small generating sets has more than half a century-long history. Indeed, this topic goes back (at least) to Gelfand an Ponomarev [7]; see Zadori [14] for details of their result on subspace lattices. For small generating sets in some other lattices, see also the introductions and the bibliographic sections of Czedli [1-3]. Recently in [1] and [3], we have pointed out that large lattices and large powers of (small) lattices can have applications in cryptography provided that they have small generating sets. This led to the original motivation of the present paper: we wanted to determine how many elements are needed to generate a large direct power of a small distributive lattice.

Even though we prove only estimates rather than exact Sperner theorems in Section 4, they are sufficient to determine the minimum number of generators of direct powers of the corresponding distributive lattices with quite good accuracy and, in most of cases, exactly; this will be formulated in (4.7) and exemplified explicitly by (4.15) and implicitly by all collections of concrete data displayed in the paper. Note that even less accuracy would be satisfactory from a cryptographic point of view, in which the role of a small minimum number of generators is to indicate that there are many small generating sets. Hence, in addition to the exact lattice theoretical results that we can obtain by combining Theorem 1 with Observation 1 or (2.12), Section 4 also offers new possibilities for the cryptographic protocols given in [1] and [3].

The purpose of this section is to generalize a result of Czedli [3] from finite Boolean lattices to finite direct powers of finite distributive lattices. To do so, we are going to borrow several concepts, notations, and ideas from [3] without further notice. Except for the sets N+ := {1,2,3,... } and No := {0} U N+, all sets and structures in this paper are assumed to be finite even when this is not explicitly mentioned.

Next, we recall some concepts and notations, and introduce a few new ones. For a real number x, the lower integer part and the upper integer part of x are denoted by |_xj and |"x], respectively. For n € N0, note the rule: |_n/2j + |~n/2] = n. A function f: N0 ^ N0 is non-bounded if for each k € N0, there exists an n € N0 such that f (n) > k. For a non-bounded function f: N0 ^ N0, the left adjoint f * of f is the function

(The terminology "left adjoint", taken from Czedli [2], is explained by categorified posets, but we do not need this fact.) If f(x) < f (y) holds whenever x < y, then f is an increasing function. For N0 ^ N0 functions f and f2, fi < f2 means that fi(x) < f2(x) holds for every x € N0. The following lemma follows straightforwardly from definitions and it belongs to folklore, so we do not prove it in the paper.

2. A bridge between combinatorics and lattice theory

f *: N0 ^ N0 defined by k ^ min{n € N0 : k < f (n)}.

(2.1)

Lemma 1. If f, fi; and f2 are increasing non-bounded N0 — N0 functions then so are their left adjoints. Furthermore, for all n, k € N0,

k < f (n) if and only if f *(k) < n, (2.2)

k > f (n) if and only if f *(k) > n, (2.3)

f (n) = max{y € N0 : f *(y) < n}, and (2.4)

fi < f2 if and only if f* < f*. (2.5)

and a natural number k € N+, let kU = (kU, <) denote the cardinal sum of k of U. That is, if (Ui; pi), ..., (Uk; pk) are pairwise disjoint isomorphic copies of

(kU; <) := (Ui U ■ ■ ■ U Ufc; pi U ■ ■ ■ U pfc).

Then for x € U¿ and y € Uj, if i = j, then neither x < y nor y < x, that is, x and y are incomparable, in notation, x y y. In other words, U¿ and Uj are unrelated for i = j. We obtain the (Hasse) diagram of kU by putting k copies of the diagram of U side by side. For k € No, the (k + 1)-element chain will be denoted by Ck. Note that kC0 is the k-element antichain. For n € N+, [n] will stand for the set {1,... ,n} while [0] := 0. For a set A, the powerset lattice (also called the subset lattice) of A is the lattice ({X : X C A}; C). In this lattice, which we denote by P(A) or (P(A); C), the operations V and A are U and n, respectively. For an element y in a poset U, we denote {x € U : x < y} by ly or, if confusion threatens, by lu y. Similarly, ty and tu y stand for {x € U : y < x}.

For posets U1 and U2 and a function ^: U1 ^ U2, ^ is an order embedding if for all x,y € U1,

x < y ^^ ^(x) < ^(y).

Let : U1 U2 denote that ^ is an order embedding. Furthermore, let U1 exists U2 denote that there exists an order embedding ^: U1 ^ U2. For example, if U is a poset, then the function U ^ P(U) defined by y ^ luy is an order embedding. Thus,

for any poset U, we have that U eXÜts P([|U|]). (2.6)

If U1 C U2 and the function U1 ^ U2 defined by x ^ x is an order-embedding, then U1 is a subposet of U2; this fact is denoted by U1 < U2. A poset cannot be empty by definition; the only exception is that for every poset U, 0U is a subposet of (and is embedded into) any other poset; the following definition needs this convention.

Definition 1. Let U be a finite poset. For k,n € N0; let

S(U, n) := max{k € No : kU eXÍ$s P([n])} and (2.7)

S*(U, k) := min{n € No : kU eXÍ$s P([n])} = min{n € No : k < S(U,n)}; (2.8)

(2.6) implies that the definition ":=" in (2.8) makes sense. For n € N+, let

fsb(n) := (^j) and fSb(k) := min{n € N+ : k < fsb(n)}. (2.9)

For the sake of better outlook and optical readability, let us agree that in in-line formulas, we often write Cbin(m,t) instead of (m); especially when m or t is a complicated expression with subscripts. With this convention, fsb(n) = Cbin(n, |_n/2j).

For a poset U isomorphic copies U = (U; <), then

Remark 1. The notation in Definition 1 is coherent with (2.1) since the functions S*(U, —): N0 ^ N0 defined by k ^ S*(U, k) and /s*b are the left adjoints of the functions S(U, -): N0 ^ N0 defined by n ^ S(U, n) and /sb, respectively. (For S*(U, -), this follows immediately from kU P([n]) ^ k < S(U, n).)

The remark above enables us to benefit from Lemma 1. Note that the notation /sb comes from Sperner's original Binomial coefficient as a Function. For subsets X and Y of [??.], using the terminology of Griggs, Stahl, and Trotter [9], we say that X and Y are unrelated if x || y for all x € X and y € Y. So S(U, n) is the maximum number of pairwise unrelated isomorphic copies of U in P([n]).

With the notation introduced in Definition 1, Sperner's Theorem from [13] asserts that S(C0,n) = /sb(n) while a Sperner theorem (i.e., a Sperner-type theorem) proved by Griggs, Stahl and Trotter [9, Theorem 2] asserts that

for t € N+, S(Ct,n) = /sb(n - t), that is, S(Q,n) = (^J-^) • (2.10)

Note that, by convention, /sb(n — t) = 0 for n < t. For later reference, some values of S(C4,n) are as follows; here and later: the numbers of our tables in exponential forms are approximations in which the significands are correctly rounded to the given digits

n 17 18 2024 2025 2026

S(C4,n) 1716 3 432 2.137- 10606 4.272 • 10606 8.544 • 10606

(2.11)

The length of a finite poset U is the largest t such that Ct is a subposet of U. The result cited in (2.10) has been generalized by Katona and Nagy [10, Theorem 4.3] to the following one.

If U is a finite poset of length t such that S*(U, 1) = t then,

for every n € N0, S(U, n) = /sb(n — t). (2.12)

A proper sublattice of a lattice L is a nonempty subset X of L such that X = L and X is closed with respect to V and A. A subset Y of L is a generating set of L if no proper sublattice of L includes Y. As L is assumed to be finite, the least size of a generating set of L makes sense; we denote it by

Gmin(L) := min{|Y| : Y is a generating set of L}. (2.13)

In the k-th direct power Lk := L x ■ ■ ■ x L (k-fold direct product) of L, the lattice operations are performed component-wise; we are interested in Gmin(Lk) for some distributive lattices L. The set of join-irreducible elements of L is denoted by J(L); by definition, x € L belongs to J(L) if and only if x covers exactly one element; in particular, the smallest element 0 = 0l of L is not in J(L). With the order inherited from L, J(L) = (J(L); <) is a poset.

Now that we have (2.13) and Definition 1, we can formulate the main result of the paper.

Theorem 1. If D is a finite distributive lattice and 2 < k € N+, then Gmin(Dk) = S*(J (D), k).

Proof. We are going to use lots of ideas from Czedli [3], where the theorem was proved for the particular case when D is a finite Boolean lattice.

For t € N+, denote by Fmeet (t) = Fmeet (x1,..., xt) the free meet-semilattice with free generators x1, ..., xt. We know from folklore and from §4 in Page 240 of McKenzie, McNulty and Taylor [12] (and it is not hard to see) that Fmeet(t) is a subposet of P([t]); in fact, Fmeet(t) is (order isomorphic to) P([t]) \{[t]}.

Let U := J(D). With U1 := U x {0} x ■ ■■x {0}, ..., Uk := {0} x ■ ■ ■ x {0} x U, it is clear that U1 U ■ ■ ■ U Uk C J(Dk). As each element x of Dk is the join of some elements of U1 U ■ ■ ■ U Uk, we have that J(Dk) = U1 U ■ ■ ■ U Uk ^ kU.

To prove that Gmin(Dfc) > S*(J(D),k), let n := Gmin(Dk) and pick an n-element generating set {gi,... , gn} of Dk. By (2.2), we need to show that k < S(U, n). So, we need to embed kU into P([n]). As Fmeet(n) = Fmeet(x1,..., xn) is embedded into P([n]) and kU = J(Dk), it suffices to give an order embedding J(Dk) — Fmeet(n). In the meet-semilattice reduct (Dk; A) of the lattice (Dk; A, V), let B := [g1,... , gn]A denote the meet-subsemilattice generated by {g1,... , gn}. By the distributivity of the lattice Dk, each u € J(Dk) is obtained so that we apply a disjunctive normal form to the generators g1,... , gn. That is, u is the join of some meets of the generators. By the join-irreducibility of u, the join is superfluous, and so u is the meet of some of the g1,... , gn. Thus, u € B, and we have seen that J(Dk) C B. Since Fmeet(n) is free, there exists a (unique) meet homomorphism : Fmeet(n) — B such that ^(xj) = gj for all i € {1,... ,n}. Since each of the generators gj of B is a ^-image, is surjective. Define a function ^: B — Fmeet(n) by the rule

^(b) := A{p € Fmeet(n) : <^(p) = b}.

Then, for every b € B,

^C0(b)) = {p € Fmeet(n) : <^(p) = b}) = /\{^(p) € Fmeet(n) : <^(p) = b} = b

shows that <p(0(b)) = b. Hence, ^(b) is the least preimage of b with respect to <p. Now assume that b1, b2 € B. If b1 < b2, then

^(b1) A ^(b*)) = <pC0(b1)) A ^(b*)) = b1 A b* = b1

shows that ^(b1) A ^(b2) is a ^-preimage of b1. As ^(b1) is the smallest preimage, we obtain that ^(b1) < ^(b1) A ^(b2) < ^(b2), that is, ^ is order-preserving. Conversely, if ^(b1) < ^(b2), then

b1 = <pC0(b1)) = ^(^(b1) A ^(b*)) = ^(^(b1)) A ^(b*)) = b1 A b* < b*,

whereby ^: B — Fmeet(n) is an order-embedding. Restricting ^ to J(Dk), we obtain an embedding of J(Dk) into Fmeet(n), as required. Consequently, Gmin(Dk) > S*(J(D),k).

To prove the converse inequality, Gmin(Dk) < S*(J(D), k), now we change the meaning of n as follows: let n := S*(J(D),k). We have to show that Dk has an at most n-element generating set.

Let U := J(D); then kU = J(Dk) as in the first part of the proof. Furthermore, we know from (2.8) that kU is order embedded in P([n]). Since k > 2, kU has no largest element. Thus, using that Fmeet(n) is order isomorphic to P([n]) \ {[n]}, kU is also embedded in Fmeet(n) = Fmeet(x1,..., xn). So we assume that kU is a subposet of Fmeet (n). A subset X of kU is called a down-set of kU if for every y € X, y C X. The collection Dn(kU) = (Dn(kU); C) of all down-sets of kU is a distributive lattice. Since kU = J(Dk), we obtain by the well-known structure theorem of finite distributive lattices, see Gratzer [8, Theorem 107] for example, that Dn(kU) = Dk. Hence, it suffices to find an (at most) n-element generating set of Dn(kU). For i € {1,...,n}, define Y := {y € kU : y < xj, understood in Fmeet(n)}. Then Y € Dn(kU), and we are going to show that {Y1,..., } generates Dn(kU). For every X € Dn(kU), X = (Jy : y € X} = V{Wy : y € X}. Therefore (since the meet in Dn(kU) is the intersection), it suffices to show that for each u € kU,

Wy = p|{Y : u € Y}.

The "C" inclusion here is trivial since the Y's are down-sets. To verify the converse inclusion, assume that v € P|{Y : u € Yj}. This means that for all i € {1,..., n}, if u € Y, then v € Y. In other words, for all i € {1,...,n}, if u < xj, then v < xj. Thus, v < A{xj : u < Xj}. As each element of Fmeet (n) is the meet of all elements above itself, u = /\{xj : u < xj}. By this equality and the just-obtained inequality, v < u, that is, v € u. This shows the "5" inclusion and completes the proof. □

3. A Sperner type theorem

Let us repeat that a poset U is bounded if 0 = 0U € U and 1 = 1U € U. Even though we have not seen the following statement in the literature, all the tools needed in its proof are present in Lubell [11], Griggs, Stahl, and Trotter [9], and Dove and Griggs [6]; this is why we call it an observation rather than a theorem.

Observation 1. Let U be a finite poset, let n, k € N0, and let p := S*(U, 1), that is,

p = min{p' € N0 : U eqts P([p'])}.

Then the following four assertions hold.

(a) If n > p, then S(U, n) > /sb(n — p).

(b) If k > 1, then S*(U, k) < p + /s*b(k).

(c) If U is bounded and n > p, then S(U,n) = /sb(n — p), i.e., S(U,n) = Cbin(n — p, |_(n — p)/2_|).

(d) If U is bounded and k > 1, then S*(U, k) = p + /s*b(k).

If |U| = 1, then p = 0. Hence, Sperner's Theorem, see [13], is a particular case of Theorem 1. Clearly, so is (2.10), which we quoted from Griggs, Stahl and Trotter [9]. The forthcoming Table 1 shows that parts (c) and (d) would fail without assuming that U is bounded.

Proof. As we have already mentioned, all the ideas are taken from Lubell [11], Griggs, Stahl, and Trotter [9], and Dove and Griggs [6].

To prove part (a), let B := {n — p + 1,n — p + 2,... ,n}. As |B| = p and we can replace U with a poset isomorphic to it, we assume that U C P(B). The |_(n—p)/2j-element subsets of {1,..., n—p} form a k := /sb(n — p)-element antichain $ in P([n — p]). For X1;X2 € $ and Y[,Y2 € U, if X1 = X2, then some i € {1,... , n — p} is in X1 \ X2 and so i € (X1 U Y1) \ (X2 U Y2). Hence, ({XUY : X € $ and Y € U}; C) = (kU; <) is a subposet of P([n]). Thus, S(U, n) > k = /sb(n—p), as required.

To prove part (b), observe that for k > 1, part (a) implies that

{n : p < n € N0 and k < S(U,n)} 5 {n : p < n € N0 and k < /sb(n — p)}. (3.1)

Observe also that, by (2.8), k < S(U,n) ^^ kU P([n]). Hence, we can compute as follows; note that (3.1) will be used only once

S *(U,k) (=8) min{n : n € N0 and k < S(U, n)} (3.2)

k> 1

= min{n : p < n € N0 and k < S(U, n)} (3.3) (3.1)

< min{n : p < n € N0 and k < /sb(n — p)} (3.4)

= min{p + n' : n' € N0 and k < /sb(n')} (3.5)

= p + min{n' : n' € N0 and k < /sb(n')} = p + /s*b(k). (3.6)

To prove (c), assume that U is bounded. It suffices to verify that

S(U,n) < /sb(n — p),

which is the converse of the inequality proved for part (a). With the notation k := S(U,n), we know that there exists an order embedding /: kU ^ P([n]). Let U1, ..., Uk be the pairwise disjoint

isomorphic copies of U such that kU is the union of them. For i € [k], denote the restriction of / to Uj by /¿, and let Xj := /i(1Ui) and Zj := /i(0Ui). Since the interval

[Zi,Xi] = {Y € P([n]) : Zj C Y C Xj}

is order isomorphic to P(Xj \ Zj), it follows that |Xj \ Zj| > p. Hence, we can pick a chain Zj = Y0(j) C Y1(j) C ■ ■ ■ C Y— C Yp(j) = Xj. If we had that Ys(j) C Yt(j) for some i = j € [k] and s, t € {0,... ,p}, then

/(0Ui) = /j(0Ui) = Zj = Y0(j) C Y(j) C Yt(j) C Yp(j) = Xj = /(1u,) = /(1U,)

and the fact that f is an order embedding would imply that 0U¿ < , which is a contradiction. Hence YS(i) and are incomparable for i = j. Therefore, letting

kCp = U

ie[fc]

with y(i) ■ ■ ■ the "capitalizing map" kCp ^ P([n]) defined by y(i) ^ Ys(i) is an order

embedding. Thus, it follows from Griggs, Stahl, and Trotter's result, quoted here in (2.10), that

S(U,n) = k < S(Cp,n) = fsb(n - p),

as required. We have shown part (c).

To prove part (d), observe that in the argument for (b), part (a) yielded inequality (3.1), which was used only once in (3.2)-(3.6). Now that part (c) turns (3.1) into an equality, (3.2)-(3.6) turn into a computation proving the required equality S *(U, k) = p + fs*b(k), completing the proof. □

4. Lower and upper estimates for non-bounded posets

For any finite poset U, Dove and Griggs [6] and Katona and Nagy [10], independently from each other, gave lower estimates and upper estimates of S(U, n). Their estimates are asymptotically equal if n tends to infinity. Thus, S(U, n) is asymptotically known3 for each U. In general, however, this knowledge does not give us too much information on S(U, n) for a small n. By parsing the arguments in Dove and Griggs [6] or Katona and Nagy [10], one can obtain some estimates for a small n but sometimes, putting generality aside, other constructions could be easier and could give better estimates. This will be exemplified by two small concrete posets; see Propositions 1 and 2 later. But first of all, let us agree that the set of all permutations of [n] are denoted by Symn; its members are written in the form n = (n1,..., nn). For n € Symn, j € [n] and X € P([n]) \ {0}, we denote by

Is(j, n) := {nm : 1 < m < j}, Lp(X, n) := max{m € [n] : € X}, (4.1)

and r(X) := {n € Symn : Is(Lp(Zj, n), n) C X} (4.2)

the j-th initial set of n, the last position of X in n, and the set of permutations associated with X, respectively. We let Lp(0,n) := 0 and Is(0, n) = 0. Of course, we can change "C" in (4.2) into "=".

3When writing arXiv:2308.15625v2, the earlier version of this paper, I did not know about Dove and Griggs [6] and Katona and Nagy [10]; thank goes to Daniel Nagy (the second author of [10]) to call my attention to these two papers.

The following statement is due to Lubell [11] and (apart from terminological changes) was used successfully by Dove and Griggs [6], Griggs, Stahl, and Trotter [9], and Katona and Nagy [10]:

if X, Y € P([n]) such that X || Y, then r(X) n r(Y) = 0 and (4.3)

for every X € P([n]), we have that |r(X)| = |X|! ■ (n - |X|)!. (4.4)

Next, let W denote the 4-element poset W with 0 and three maximal elements, see Fig. 1. For n € N+ we define

n

^»»"li^TO'M»-!)]. (4.5)

With the convention that Cbin(ni,n2) = 0 unless 0 < n2 < ni, let

n — 3i — 3 j VL(n — 1)/2j + j — 3i

( Ln/3J-1

ioS(W,n) :=

E EC)L-ni-/2i;--J if ni{з,^7},

i=0 j=0 Ln/3J-1 i ,.N ,

y y 3j % n - 3 ) if n G{3, 5, 7}. i=0 ¿0 W V(n - 3)/2+ j - 32^ 1 1

(4.6)

Note that ioS(W, 1) = S(W, 1) and loS(W, 2) = S(W, 2). Hence, we can often assume that n > 3. The natural density of a subset X of N+ is defined to be |X n [n]|/n, provided that this

limit exists.

Proposition 1. For 3 < n € N+, upS(W, n) and loS(W, n) defined in (4.5) and (4.6) are an upper estimate and a lower estimate of S(W, n), that is,

loS(W, n) < S(W, n) < upS(W, n).

The functions loS(W, —), S(W, —), upS(W, —), and 1/4 ■ /sb(—) are asymptotically equal. Furthermore, denoting the left adjoints of the functions loS(W, —) and upS(W, —) by loS*(W, —) and upS*(W, —), respectively,

upS * (W, k) < S * (W, k) < loS * (W, k) and 0 < loS * (W, k) — upS * (W, k) < 1 (4.7)

for all k € N+, and the natural density of the set

{k € N+ : upS*(W, k) = loS*(W, k)}

is 1 .

The proof below uses lots from the proofs in Dove and Griggs [6] and Katona and Nagy [10]; we are going to discuss the differences in Remark 2.

Proof. First, we deal with upS(W, n). Let k := S(W, n), and let W1,..., Wk be pairwise unrelated copies of W in P([n]). In particular, (Wj, C) is order isomorphic to W. The assumption n > 3 gives that upS(W, n) > 1. Thus, we can assume that k > 2 as otherwise

S(W, n) = k < upS(W, n)

is obvious. In accordance with Figure 1, we use the notation Wj = {Zj,Cj,Dj, E} where Z C Cj, Cj || Dj, etc., and Zj || Ej for i = j, etc. As it is trivial (and used also in Dove and Griggs [6] and Katona and Nagy [10]), if i = j € [k], Y € P([n]), Y', Y'' € Wj, and Y' C Y C Y'', then Wj U {Y}

is still unrelated to W?; we are going to use this "convexity principle" implicitly. As its first use, we can assume that Zj equals the intersection Cj n Dj n Ej as otherwise we could replace Zj by this intersection.

We claim that with some pairwise distinct elements cj,dj,ej € [n] \ Zj, we can change Wj to Wj = {Zj, Zj U {cj}, Zj U {dj}, Zj U {ej}} such that Wb ..., Wj_b W/, Wj+1, ..., Wk still form a system of pairwise unrelated copies of W. Let Cj = Cj \ Zj, Dj = Dj \ Zj, and Ej = Ej \ Zj. If at least one of Cj, Dj and Ej is not a subset of the union of the other two, say, Cj ^ D' U Ej, then any choice of cj € Cj \ (D' U Ej), dj € D' \ Ej, and ej € Ej \ Dj does the job by the convexity principle. So we can assume that each of Cj, D' and Ej is a subset of the union of the other two. Take an element from Cj \ D'. As Cj C Dj U Ej, this element is in Ej; we denote it by £c,-.d,e . The meaning of its subscripts is that xc,-d,e belongs to Cj and Ej but not to D'. By symmetry, we obtain elements xc,d,-e € (CjnDj) \Ej and x-c,d,e € (Dj nEj) \Cj. The subscripts show that these three elements are pairwise distinct. This fact and the convexity principle imply that Wj can be changed to the required form with cj := xc,-d,e, dj := xc,d,-e, and ej := x-c,d,e. Therefore, in the rest of the proof, we assume that for all i € [k],

Wj = {Zj, Zj U {c}, Zj U {dj}, Zj U {e}}.

Letting

r := r(Zj) U r(Zj U {c}) U r(Zj U {dj}) U r(Zj U {e}),

our next task is to find a reasonable lower bound on |rj|. With the notation zj := |Zj|, we can order the first zj components of a

n = (n!,...,nra) € r(Zj) n r(Zj U{Cj}),

which form the set Zj, in z! ways. We have that nZi+1 = cj, and the last n — zj — 1 components can be ordered in (n — zj — 1)! ways. Hence,

|r(Zj U{c})| = z!(n — z — 1)!,

and the same is true for |r(Zj U {dj})| and |r(Zj U {ej})|. This fact, (4.3), (4.4), and the inclusion-exclusion principle yield that

|r | = g0(zj), where g0(x) := x!(n — x)! + 3(x + 1)!(n — x — 1)! — 3x!(n — x — 1)! (4.8)

= (n + 2xj)xj!(n — 1 — Xj)!.

Note that zj > 1 as otherwise Zj = 0 would be comparable with Zj for j € [k] \ {i}. (Here we used that k > 2.) We also have that zj < n — 1 since Zj U {cj} € P([n]). Thus, we can use later that x € [n — 1] = {1,..., n — 1}. For the auxiliary function

01 (x) := g0(x) — g0(x — 1),

we have that

g1(x) = g2(x) ■ x!(n — 1 — x)!, where g2(x) = 4x2 — 2x — (n2 — 2n).

The smaller root of the quadratic equation g2(x) = 0 is negative while the larger one is strictly between n/2 — 1/2 and n/2 — 1/4. Hence the largest integer x for which g2(x) and so g1(x) are negative is x = |_(n — 1)/2j. Therefore, on the set [n — 1], g0 takes its minimum at |_(n — 1)/2_|.

A

(0)

r

b4

b3 b2 bi bo

T

a(1) a

I \ I \ I I 1-^

i i

T

i'2'j

(2)

/ ^

/ \ / 1 i i 1-

T

ï3

a(3) a(4) a(°) a aaaa

22

T

2

2

(4)

23

(5)

T

3

2

32

I ' 1 '

T

2

3

(6) 33

T

3

3

a(7) a(°) a(9) a

a 000 aoqq aooq a

222

T

2

2

2

,(°)

322

(9)

T

2

2

3

232

T

2

3

2

.(10) 223

T

3

2

2

Figure 2. Copies of A := [16]. Here 1 = • • subset. In each other copies, the total number of elements in the ovals is also 7.

|B4| = 3. In the copy A0), the oval stands for a 7-element

Let M := g0(|_(n — 1)/2j). Using that the r»'s are pairwise disjoint by (4.3) and r U ... rk C Symn, we obtain that

kM = y M <¿>11 < |Sym„| = n!.

(4.9)

i=1

i=1

Dividing this inequality by M (and dealing with odd n's and even n'-s separately), we obtain the required inequality

S(W,n) = k < upS(W,n).

Next, we turn our attention to ioS(W, n). Let m := |_n/3j. For n/ {3, 5, 7}, let h := |_(n —1)/2j. For n € {3, 5, 7}, h stands for (n — 3)/2. With A := [n], let us fix pairwise disjoint 3-element subsets B0, Bi, ..., Bm-1 of A, and denote the "remainder set" A\ (B0 U • • • U Bm-1) by R. These subsets are visualized in Figure 2, where n = 16, m = 5, h = 7, and A with subscripts and superscripts is drawn eleven times (in four groups separated by spaces). We can assume that n > 3. For i € {0,..., m — 1}, the elements of B» are denoted as follow: B» = {c», d, e»}. For i € N0, call a vector v = (v0,..., vi-1) € {2,3}» eligible if i < m — 1 and v0 + v1 + • • • + vi-1 < h. Note that for i = 0, the empty vector is denoted by 0 and it is eligible. As Figure 2 shows, there are exactly eleven eligible vectors for n = 16; they are the lower subscripts of the copies of A; because of space consideration, we write 232 instead of (2,3,2), etc., in the figure. (The upper subscripts of A help to count the copies but play no other role.)

For each eligible v, we define a family of copies of W in P([n]) = P(A) as follows. Let i denote the dimension of v, that is, v = (v0,..., vi-1). For j = 0,..., i — 1, pick a Vj-element subset Xj of Bj. In the figure, Xj is denoted by a dotted oval with Vj sitting in its middle. Furthermore, pick a subset X» of A \ (B0 U B1 U • • • U B») such that X» = h — v0 — • • • — vi-1. In the figure, X» is the dashed oval (without any number in its middle). Let us emphasize that Xj C Bj holds only for j < i but it never holds for j = i. Denote (X0,X1,..., X») by X, call it an eligible set vector, and

Clearly,

let Z^ := X0 U ■ ■ ■ U X

regardless the choice of v and X, we have that |Zjx | is always the same, namely, | = h.

(4.10)

For convenience, let Cj := {ci}, di := {di}, e-i := {e-i}, and Zi := 0. Observe that {ci, di, e-i, Zi} is a copy of W in P(Bj); in each copy of A in the figure, this copy of W is indicated by its diagram for

3

exactly one i. It follows that

is also a copy of W but now in P(A) = P([n]).

To prove that ¡oS(W, n) < S(W, n), we need to show that ioS(W, n) is the number of eligible set vectors X and for distinct eligible set vectors X = X*, the corresponding copies W^ and W^• of W are unrelated.

First, we deal with the number of eligible set vectors XX = (X0,..., Xj). As each of the Bj's are 3-element and there are |_n/3j many of them, the largest value of i is at most |_n/3j — 1, the upper limit of the outer summation index in (4.6). The eligible vector v that gives rise to XX is uniquely determined by X since v = (|X01,..., |Xj_1|).

Let j := |{t € {0,..., i — 1} : vt = 2}|. This j, which corresponds to the inner summation index in (4.6), is the number of 2's in dotted ovals in the figure. There are (j) possibilities to choose the j-element set {t € {0,..., i — 1} : vt = 2}; this is where the first binomial coefficient enters into (4.6). For each t € {0,..., i — 1} such that vt = 2, we can choose the 2-element subset Xt of Bt in 3 ways. As there are j such t's, this brings the power 3j into (4.6). Since Xj is a subset of the n — 3i — 3-element set A \ (B0 U ■ ■ ■ U Bj) and

|Xj| = h — V0-----vj_1 = h — 2j — 3(i — j) = h + j — 3i,

the second binomial coefficient in (4.6) gives how many ways we can choose Xj. Therefore, (4.6) precisely gives the number of eligible set vectors X.

Next, assume that X = (X0,...,Xj) and X • = (XJ,..., X*) are distinct eligible set vectors with corresponding (not necessarily different) eligible vectors v = (v0,..., vj—1) and v* = (vJ,..., v*V_1). Assume also that a and a* are in W such that (X, a) = (X*, a*). We need to show that a^ = Hi U Zx and =a* U Zare incomparable. There are two cases to consider; both can easily be followed by keeping an eye on Fig. 2 in addition to the formal argument.

First, assume that i = i*, say, i < i*. Observe that |aX• n Bj| = |X*| = v* > 2 but |a^ n Bj| = \a,i\ < 1. So n Bi\ > fl (Pictorially, a dotted oval, labeled by 2 or 3, has more elements than \a,i\ symbolized by one of the vertices of the diagram of W drawn in Bj.) Hence, aX• ^ ax. For the sake of contradiction, suppose that a^ C aX•. Then for every j € {0,..., i — 1}, vj = |Bj nax| < |Bj n aX• | = v*. Hence, we can compute as follows; the computation is motivated

by comparing, say, a21) and a232 in Fig. 2:

|ax n (Bj+1 U ■ ■ ■ U Bm_1 U R)| = |Xj| = h — v0-----vj_1 > h — v0-----v*_1

= (h -Vq-----<._!) + (v*+1 + • • • + <._!) + |a*. | + (v* - |a*. |)

= \x'< I + i\x'+i \ + ''' + \x'--i\ + + (v' ~ I a*. |)

= |a^. n {Bi+1 U • • • U Bm_i UR)\ + « - |a*. |) > n {Bi+1 U • • • U Bm_i Ufl)|.

The strict inequality just obtained contradicts that a^ C aX•, and we conclude in the first case that

a^X || axx •, as

required.

Second, assume that i = i*. If Xj || X* for some j € {0,..., i} or there are s, t € {0,..., i} such that4 Xs C X* but Xt D Xt*, then the validity of a^ || a*^ is clear. Thus, we can assume

X xX •

that Xj C Xj* for all j € {0, . . . , i}. Then

h = |X0| + ■ ■ ■ + |Xj| < |X*| + ■ ■ ■ + |Xj*| = h

4 According to the convention of lattice theory, "c" is the conjunction of "Ç" and "=".

Table 1. Some values of ioS(W, n) and upS(W, n); the known values of S(W,n) are encircled.

n 3 4 5 6 7 8 9 10 11 12 13 14

lo S(W,n) ® ® ® © 9 17 36 66 120 234 456 876

up5(IU, n) 1 2 3 6 10 20 37 70 132 252 480 924

n 15 16 17 18 19 20 21

io S(W,n) 1680 3 625 6 340 12 330 23 960 46 766 91224

upS(IU, n) 1775 3 432 6 630 12 870 24 967 48 620 94631

n 22 23 24 25 26

io S(W,n) 178 388 348 656 683130 1337896 2 625 364

apS(W., n) 184 756 360 554 705 432 1 379 671 2 704156

n 27 28 29 30

io S{W,n) 5149 872 10119 348 19 877904 39104 856

apS(W., n) 5 298 418 10 400 600 20 410 200 40116 600

together with |Xj1 < |X*|, for all j € {0, ...,i}, imply that Xj = X* for all j € {0, ...,i}. Combining this equality with Xj C Xj for all j € {0,..., i}, we obtain that X = Xcontradicting our assumption. We have shown that ioS(W, n) < S(W, n), as required. It is well known that no matter how we fix two integers s and t,

(n_s \ l n \

Ln/2J - J is asymptotically 2-si |n/2j) = 2-s/sb(n) if n ^ to; (4.11)

this folkloric (and trivial) fact was used in Dove and Griggs [6] and Katona and Nagy [10], too. This fact and (4.5) yield that upS(W, _) is asymptotically 1/4 ■ /sb(_). Hence, to obtain the required asymptotic equations, it suffices to show that loS(W, _) is asymptotically 1/4 ■ /sb(_), too. Let n and p be small positive real numbers. As ^°=0 2-i = 2, we can fix a q € N+ such that ^q=0 2-i > 2 _ n. Using (4.11) and assuming that i < q, we obtain that the second binomial coefficient in (4.6) is asymptotically 2-3i/sb(n _ 3) or, rather, it is 1/8 ■ 2-3i/sb(n). So it is at least 1/8 ■ 8-i(1 _ p)/sb(n) for all but finitely many n. Hence, assuming that n is large enough and, in particular, |_n/3j > q,

q i 1 1 q

lo o{yv,n, ^ °"\j) ' 0 • 8(1 - At)/sb(n) = 8(1 " At)/sb(n) 'El;1'"'1

i=0 j=0 j i=0 j=0 j (4 12)

") ^ E E ( •) • - /0/sb(n) = kl- n)U(n) £ 8- £ (') y • l4-

i=0 j=0 ^^ i=0 j=0

- p)/sb(») E 8-(3 + ir > - p)/sb(»)(2 - r?) =

8 i=0 8 8

As the last fraction in (4.12) can be arbitrarily close to 1/4, it follows that loS(W, n) is asymptotically at least 1/4 ■ /sb(n). It is asymptotically at most 1/4 ■ /sb (n) since so is upS(W, n) and we know that loS(W, n) < S(W, n) < upS(W, n). This completes the argument proving the "asymptotically equal" part of Proposition 1.

Next, we turn our attention to the left adjoints of our estimates. First of all, we claim that

for every n € N+, upS(W, n) < loS(W, n + 1). (4.13)

Let up+S(W, —) be the same as upS(W, —) except that we drop the outer "lower integer part" function from its definition. It suffices to prove (4.13) with up+S(W, n + 1) instead of upS(W, n + 1). We can assume that n > 10 as otherwise (4.13) is clear by Table 15. Let T(n) denote the sum of the two summands in the upper line of (4.6) that correspond to (i,j) = (0,0) and (i,j) = (1,1). After a straightforward but tedious calculation, if n = 2m, then

up+ff(IE, n) up+S(W,ra) _ 2m(2m - 2)(2m - 3)

i0S(W, n + 1) - T(n + 1) (m - l)2(llm - 12) ' '

Subtracting the numerator from the denominator, we obtain 3m3 — 14m2 + 23m — 12, which is clearly nonnegative for 5 < m € N+ (in fact, for all m € N+), whence the fraction is at most 1 for n = 2m > 10. For an odd n = 2n + 1 > 10, (4.14) turns into

up+ff(IE, n) up+ff(IE, n) _ 4(2m + l)(2m-l)(2m-3) i0S{W,n + l) ~ T(n + 1) ~ (4m + l)(llm2 - 19m + 6) '

and now the subtraction gives the polynomial 12m3 — 17m2 + 13m — 6, which is clearly nonnegative for 2 < m € N+ (in fact, for all m € N+). Thus, passing from m to n, the required inequality up+S(W,n) < loS(W,n + 1) holds for all 10 < n € N+. We have shown the validity of (4.13).

Next, we deal with (4.7). By Table 1, the first few values of upS *(W, k) and those of loS *(W, k) are as follows:

k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

upS*(W,k) 3 4 5 6 6 6 7 7 7 7 8 8 8 8 8

io S*(W,j) 3 5 6 6 6 6 7 7 7 8 8 8 8 8 8

(4.15)

This implies (4.7) for k < 15 (in fact, for k < 29), so we can assume that k > 15. Using (4.15) and the obvious fact that loS(W, —) is a strictly increasing function on N+ \ [7], there is a unique 7 < n € N+ such that

ioS(W,n) < k < loS(W,n + 1). Using (4.13) and the inequality loS(W, n) < upS(W, n), we obtain that

either loS(W,n) <k < upS(W,n) (4.16)

or upS(W,n) < k < loS(W,n + 1). (4.17)

If (4.16), then upS *(V, k) = n and *(W, k) = n + 1. If (4.17), then

upS * (W,k) = n + 1 = loS * (V, k).

In both cases, 0 < loS *(W, k) — upS *(W, k) < 1, as required. Next, for t € N+, let

Et := {k € [t] : upS *(W, k) < loS *(W,k)}.

To settle the last sentence of Proposition 1 about density, it suffices to show that limt^^(|Et|/t) = 0. Let e < 1/12 be a positive real number; we are going to show that |Et|/t < e for all but finitely many t's. Asymptotic equalities will often be denoted by As we have already proved that

5The table was obtained by the computer algebraic program Maple V Release 5, which ran on a desktop computer with AMD Ryzen 7 2700X Eight-Core Processor 3.70 GHz for 1/5 seconds.

(4.11) yields that upS(W,n — 1)/upS(W,n) ^ 1/4 as n ^ to. This fact, 1/6 < 1/4 < 3-1, and loS(W, n) ~ upS(W, n) allow us to fix an no = n0(e) € N+ such that for all n > no,

upS(W,n)/6 < upS(W,n — 1) < upS(W,n) ■ 3-1, (4.18)

upS(W, n) — loS(W, n) < upS(W, n) ■ e/12 . (4.19)

Later, it will be important that n0 does not depend on t. Hence, from now on, we can assume that upS(W, n0) < t. Since lim^^, upS(n0 + i) = to in a strictly increasing way, there exists a unique r = r(t) € N+ such that upS(n0 + r — 1) < t < upS(n0 + r). Since e is small, (4.18) and (4.19) yield that for all i € [r],

upS(W, n0 + i — 1) < loS(W, n0 + i) < upS(W, n0 + i), (4.20)

long good interval

upS(W,n0 + r)/6 <t. (4.21)

Observe that (4.20) and loS(W, n0 + i — 1) < upS(W, n0 + i — 1) imply that for every k in the left open and right closed interval (upS(W, n0 + i — 1), loS(W, n0 + i)], which is under-braced in (4.20), loS * (W, k) = upS *(W, k) = n0 + i. So this interval is disjoint from |Et| for any t € N+. Thus, letting

c := upS(W, n0 and d := upS(W, n0 + r),

Et is a subset of

[1, c] U U (loS(W, n0 + i), upS(W, n0 + i)]. ie[r]

Hence,

|Et| < c + ^ (upS(W, n0 + i) — loS(W, n0 + i))

ie[r]

(4.19) e ^ e

< c + --J2apS(W,no+i) = c + -- apS(W,no + r-i)

ie[r] ie{0,...,r-i}

(4.18) e v-^ • , ed v-^ ed 4 ed

< c + — > 3 d < c H--> 3 = c +---= c + —.

12 ^ 12 ^ 12 3 9

ie{0,...,r-1} ieNo

This inequality and (4.21) yield that

|Et|/1 < |Et|/(upS(W,n0 + r)/6) < (c + ed/9)/(d/6) = 6c/d + 2e/3.

As t ^ to, r = r(t) and d = upS(W, n0 + r) also tend to to. So for all sufficiently large t, we have that 6c/d < e/3, whereby |Et|/t < e/3 + 2e/3 = e. Thus, 0 < |Et|/t < e for all but finitely many t, and this is true for every positive e < 1/12. That is, limteN+ |Et|/t = 0. Hence, the natural density of E is 0 and that of N+ \ E, which occurs in Proposition 1, is 1, as required. The proof or Proposition 1 is complete. □

Remark 2. (Differences from [6] and [10]) The differences we are going to summarize here are partly due to the fact that, naturally, more can be proved for a small particular poset than for all finite posets. When proving that S(W, n) < upS(W, n), the only novelty is the argument between (4.8) and (4.9). More novelty occurs in our proof of loS(W,p) < S(W, n). As opposed to Dove and Griggs [6], where several "layers" are populated, we use no iteration and we have (4.10). Compared

to Katona and Nagy [10], our construction performs better for small values of n; the following table shows what lower estimates could be extracted from [10]

n 10 50 100

by [10]: 21 14 833 897 694 226 12 229 253 884 310 811 313 310 605 728

loS(W,n) : 66 31761385 392 516 25 286 044 048 404 745 303 553 386 716

(We have no similar numerical comparison in case of [6].) Except for (2.12), which is quoted from [10] and does not apply for W, [6] and [10] give only asymptotic results but no concrete values of S(U, n) for a poset U.

Remark 3. Even for a small n, the trivial algorithm for determining S(W, n) is far from being feasible. For example, for n = 10, the "cover-preserving" copies of W in P([10]) form a

7

Cbin(10, i) ■ Cbin(10 - i, 3) = 15 360-element set H.

i=0

All the (S(W, 10) + 1)-element subsets of H should be excluded, but no computer can exclude

Cbin(15 360, S(10) +1) > Cbin(15 360,67) > 10185

subsets; the first inequality here comes from Table 1.

Next, we investigate another small poset, V; see Fig. 1. Define

U(n-2)/2l/2j loS(V,n):= y i=0

2n - 3 |_n/2j - 1

n - 2 - 2i [(n - 2)/2] - 2iJ'

n-2

(4.22)

(4.23)

Proposition 2 (Mostly from Katona and Nagy [10]). For 2 < n € N+, Proposition 1 remains valid if we substitute V and 1/3 ■ /sb(-) for W and 1/4 ■ /sb(-), respectively.

A few values of loS(V, n) and upS(V, n) are listed below:

n 2 3 4 5 6 7 8 9 10 11 12 13

io S(V,n) 1 1 2 4 7 13 24 46 86 166 314 610

up5(F, n) 1 1 2 4 7 14 25 48 90 173 326 632

n 14 15 2022 2023

io S(V,n) 1163 2 269 w 2.848 220 • 10BUB w 5.695 500 • 10BUB

upS(V, n) 1201 2 340 w 2.848 846 • 10BUB w 5.696 752 • 10BUB

aps/los » 1.033 1.031 1.000 219 853 1.000 219 780

(4.24)

(4.25)

We do not prove this proposition in the paper. It would be straightforward to simplify the proof of Proposition 1 to obtain a proof of Proposition 2. (The simplification means that |B | =2 and all the eligible vectors v are of the form (1,..., 1) and so we do not need them.) Note that arXiv:2308.15625v2, the earlier version of this paper, contains a detailed proof of Proposition 2. However, our construction to prove that loS(V, n) < S(V, n) is included already in Katona and Nagy [10, last page], where loS(V,n) = S(V,n) is conjectured. Note the little typo in [10, equation (27)]; the upper limit of the summation should be |(n + 3)/2j rather than |(n + 2)/2j. After that this typo is corrected, (27) in [10] is the same as (4.22).

The computation for the following mini-table took twelve minutes; see Footnote 5

n 2022 2023 2024

loS(W,n) » 2.136194 • 10606 4.271332 • 10606 8.540 554 • 10606

upS(W, n) w 2.136 987 • 10606 4.272 916 • 10606 8.543 720 • 10606

up5/i05 » 1.000 371103 1.000 370 920 1.000 370 737

(4.26)

It follows from Propositions 1 and 2, Table 1, (2.11), (4.15), (4.24), (4.25), and (4.26) that the minimum sizes of generating sets of the k-th direct powers of the lattices Dn(V) and Dn(W), drawn in Fig. 1, and the 5-element chain C4 are given as follows

k 2022 2023 3 • 10606 5 • 10606

Cmin(C^ ) 18 18 2025 2026

Gmin(D(V)k) 15 15 2023 2023

Gmin(D(W)k) 16 16 2023 2024

5. Appendix: Maple worksheet

In this section, we present the Maple worksheet that computed Table 1; see Footnote 5. For the rest of the numerical data in the paper, either the two parameters in the "for n from 3 to 30 do" can be modified or a much simpler worksheet would do.

> restart; time0:=time():

> #An upper bound for Sp(W,n):

> upSW:= proc(n) local s; s:=n/(3*n-2-2*floor(n/2));

> floor(s*binomial(n-1, floor((n-1)/2)));

> end:

> # A lower bound for Sp(W,n):

> loSW:=proc(n) local s,i,j,ub,lb,h,summand,returnvalue;

> s:=0;

> if (n=3) or (n=5) or (n=7) then h:=floor((n-3)/2)

> else h:=floor((n-1)/2)

> fi;

> for i from 0 to ceil(n/3)-1 do ub:=n-3-3*i;

> #ub: Upper number in the 2nd Binomial coefficient

> if ub >= 0 then

> for j from 0 to i do lb:=h-2*j-3*(i-j);# j: number of 2's,

> #lb: Lower number in the 2nd Binomial coefficient

> if (lb>=0) and (lb<=ub) then

> summand:=binomial(i,j)*3"j*binomial(ub,lb);

> s:=s+summand;

> fi;#end of the "if (lb>=0) and (lb<=ub)" command

> od; #end of the j loop

> fi; #end of the "if ub >= 0" command

> od; #end of the i loop

> returnvalue:=s; #the procedure returns with the last result

> end:

> for n from 3 to 30 do lower:=loSW(n):

> upper:= upSW(n):

> if lower>0 then ratio:=evalf(upper/lower) else ratio:=undefined fi :

> print('n=', n, ' lower=' ,lower, ' upper=',

> upper, ' ratio=', ratio);

> if lower>10"6 then

> print('lg(lower)=',evalf(log[10](lower)),

> 'lg(upper)=',evalf(log[10](upper))) fi;

> od:

> time2:=time():

> print('The total computation needed time2-time0,' seconds.');

Based on Theorem 1, some results analogous to Proposition 1 have recently been proved in [4] and [5].

REFERENCES

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