The size of the maximum antichains in products of linear orders

The size of maximum antichains in the product of n linear orders is known when the n linear orders have the same length. We present an exact expression for the size of maximum antichains when the linear orders have (possibly) different lengths. From this, we derive an exact expression for the size of maximum antichains in the product of n linear orders with the same length. This expression is equivalent to but different from the existing expression. It allows us to present an asymptotic result for the size of maximum antichains of n linear orders with the same length m going to infinity.


Introduction
Antichains in the poset {0, 1} n equipped with the standard partial ordering are wellstudied and have many different interpretations (Ersek Uyanık et al. 2017). An expression for the maximal size of such antichains in {0, 1} n is given by a classical theorem due to Sperner (1928). If we consider the more general poset {1, … , m} n also equipped with the standard partial ordering, an expression for the size of maximum antichains is given by Sander (1993). Sander also provides asymptotic results when m is fixed and n goes to infinity. The interest of Sander in this problem arose from a recreational mathematics problem posed in Motek (1986). Actually, antichains and, hence, maximal antichains, in the poset {1, … , m} n are of interest in many domains. For instance in game theory, Hsiao and Raghavan (1992) define a multichoice cooperative game as a real-valued mapping on {1, … , m} n , where n is the number of players and {1, … , m} denotes the set of ordered actions that each player can take. A profile in such a game is a vector x = (x 1 , … , x n ) ∈ {1, … , m} n and represents the actions taken by each agent. A winning profile is such that the value of the game at that profile is 1. A winning profile x is minimal if there is no other winning profile y such that y ≤ x . If a game is monotone, then the set of all minimal winning profiles is an antichain. Besides, Grabisch (2016) shows that antichains in {1, … , m} n play an important role in the analysis of these multichoice cooperative games.
In Hsiao and Liao (2008), a generalization of multichoice cooperative games is presented by considering that the set of actions available to agent i is {1, … , m i } . A multichoice cooperative game thus becomes a real-valued mapping on ∏ n i=1 {1, … , m i } . Applications of multichoice cooperative games in various domains (cost allocation, voting, ...) can be found in Branzei et al. (2014); Freixas (2020). A very simple example 1 of multi-choice cooperative game is as follows. Consider a voting situation in a parliament in which the players are the political parties, m i represents the number of seats of party i, a winning profile is such that the proposal obtains enough votes to be approved, and there is no voting discipline in political parties. In this setting, minimal winning profiles are antichains and they are interesting because they give information about how to approve a proposal.
Our personal interest in antichains in the poset ∏ n i=1 {1, … , m i } stems from the analysis of a multicriteria sorting model (called ElEctrE tri-nb) presented in Fernández et al. (2017). This model, when restricted to two categories, makes use of p limiting profiles (vectors of length n) supposed to lie at the border between the upper and the lower category. An alternative x (also a vector of length n) is assigned to the upper category if x is weakly preferred to at least one limiting profile and no limiting profile is strictly preferred to x. In this context, the size of maximal antichains corresponds to the maximum number of limiting profiles needed to represent a twofold ordered partition in ElEctrE tri-nb, whenever such a representation is possible. See Bouyssou et al. (2020) for a detailed explanation of the relation between antichains and ElEctrE tri-nb.
Another paper about antichains in {1, … , m} n is Tsai (2018) (and the references therein): it presents an upper bound for the number of antichains (a generalization of Dedekind numbers).
In the present paper, we extend Sander's results in two directions. First, we present an exact expression for the size of maximum antichains in the heterogeneous product ∏ n i=1 {1, … , m i } . Then, we provide asymptotic results for the size of maximum antichains in {1, … , m} n when n is fixed and m goes to infinity.
The rest of the paper is organized as follows. In Sect. 2, we introduce some notation and the main definitions. Section 3 is devoted to the general case of heterogeneous products and presents some exact results about the size of maximum antichains. Our main results are presented in Sect. 4, in which we consider the special case of homogeneous products and we present a new exact result and also an asymptotic one when n is fixed.

Notation and definitions
Let P be a set and ≤ be a binary relation defined on P, satisfying (i) reflexivity ( ∀x ∈ P, x ≤ x ), (ii) antisymmetry ( ∀x, y ∈ P, x ≤ y and y ≤ x ⟺ x = y ) and (iii) transitivity ( ∀x, y, z ∈ P, x ≤ y and y ≤ z ⇒ x ≤ z ). The pair (P, ≤) is called a partially ordered set (poset) 2 . When there is no ambiguity, the poset (P, ≤) is simply denoted by P. For all x, y belonging to a poset P, we say that x and y are comparable if x ≤ y or y ≤ x . A chain of P is a totally ordered subset of P. A linear order on P is a poset such that P is a chain. An antichain of P is a subset of pairwise incomparable elements. A maximum antichain is an antichain of maximal cardinality.
Let (P, ≤ P ) and (Q, ≤ Q ) be two posets. The product poset (P × Q, ≤) is defined to be the set of all pairs (a, b), a ∈ P, b ∈ Q , with the order given by (a, b) ≤ (a � , b � ) if and only if (a ≤ P a � ) and (b ≤ Q b � ) . Let n be a positive integer and m = (m 1 , … , m n ) be an element of ℕ n , where ℕ denotes the set of positive integers. For any a ∈ ℕ , let [a] denote the set {1, … , a} . For any i ∈ [n] , the poset ([m i ], ≤) , where ≤ is the usual ordering of the natural numbers, is a linear order (also called a chain). The product of these n chains is the poset (Griggs 1984). When m is such that The size of the maximum antichains in ∏ n i=1 [m i ] and [m] n is respectively denoted by s(m) and S(m, n). Sperner (1928) has proved that the size of a maximum antichain in [2] n is 1 3 The size of the maximum antichains in products of linear orders When n is large, a convenient approximation for S(2, n) is obtained using Stirling's formula: S(2, n) ∼ 2 n √ 2∕ n. Later, Sander (1993) has proved that the size of a maximum antichain in [m] n is with g = ⌊n(m − 1)∕2⌋ . Sander has also provided a bound 3 and some asymptotic results for S(m, n) when m is fixed. Asymptotic results for S(m, n) when n is fixed have not been discussed in the literature. Notice that S(m, n) corresponds to Sequence A077042 in the On-line Encyclopedia of Integer Sequences (OEIS 2019).

Heterogeneous product
Since multichoice cooperative games have been generalized to heterogeneous sets of actions (Hsiao and Liao 2008) and the analysis of ElEctrE tri-nb also involves antichains in a heterogeneous product set (Bouyssou et al. 2020 Since the website only presents an informal and elliptic proof, we deem it useful to present a formal proof below. Let us first recall some definitions and results about posets. Let (P, ≤) be a poset. For any x, y ∈ P , we say that y covers x in P iff x < y and there is no z such that x < z < y . A ranking (or grading) of a poset P is a partition of P into (possibly empty) sets P i (i ∈ ℤ) such that, for each i, every element in P i is covered only by elements in P i+1 . The set P i is called the ith rank of P. If a poset admits a ranking, then we say that it is ranked (or graded). P is said to be Sperner if every rank of largest size is a maximum antichain.

Proof of Theorem 1
In this proof, for the sake of brevity, we use X to denote the poset ( , the poset ([m i ], ≤) is a chain. Hence, X is a product of chains, it is Sperner and the median rank (or ranks if n + m [n] is odd) is a maximum antichain of X (De Bruijn et al. 1951;Griggs 1984). It is simple to see that this rank is the set For any I ⊆ [n] , let us define We have D I = ⋂ i∈I D {i} and using the multiset coefficient formula (also known as 'stars and bars' technique), we find It is clear that

Using the inclusion-exclusion principle, we find
Since h − m I − 1 n − 1 = 0 whenever h − m I < n , we obtain (2). ◻ Table 1 illustrates how s(m) varies as a function of m.
In particular, we see that increasing one of the m i 's way above the others has a limited impact. When all components of m are identical, it is easy to show that (2) coincides with Sander's expression (1). In that case, Sander's expression is computationally more efficient than ours.
x i = h}.
x i = h and x i > m i ∀i ∈ I}.

3
The size of the maximum antichains in products of linear orders This section about heterogeneous products does not contain any asymptotic result because it does not seem relevant to let one of the parameters, say m 5 , go to infinity while keeping the other parameters constant.

Homogeneous product
In addition to an exact expression for S(m, n), Sander (1993) provides asymptotic results when n → ∞ . Our goal in this section is to analyze the asymptotic behavior of S(m, n) when m → ∞ . This corresponds in multichoice simple games to the case in which the number of ordered actions the players can take is very large. In the case of multicriteria sorting, this corresponds to situations in which the alternatives to be sorted are evaluated on criteria scales involving a large number of levels. To this end, we first present a new exact expression for S(m, n) from which we then derive an asymptotic result.
Let h = ⌊n(m + 1)∕2⌋ . Our first result about homogeneous products is the following.
Theorem 2 For all n ≥ 2 , if n(m + 1) is even, then S(m, n) is equal to Otherwise, S(m, n) is equal to  Proof We first prove (3). We have seen in the proof of Theorem 1 that an anti- Hence, if n(m + 1) is even, then an antichain of maximum size Since no x, y ∈ A are comparable, we know that no distinct x, y ∈ A project on the same element in  ] y i = r and y l > m} for l ∈ [n − 1]

3
The size of the maximum antichains in products of linear orders and C l * r = B r ⧵ C l r . Then A r = ⋂ l∈[n−1] C l * r and, thanks to the inclusion-exclusion principle, where the last equality holds because all dimensions play the same role. The set B r is a regular (n − 2)-dimensional simplex. Its cardinality is equal to the (r − n + 1)-th simplicial polytope number in n − 2 dimensions (Kim 2002), that is The set ⋂ l∈[i] C l r is the set of all elements of ℕ n−1 such that at least i components are strictly larger than m. If r < i(m − 1) + n − 1 , then ⋂ l∈[i] C l r is empty because it is not possible to have at least i components strictly larger than m. Hence where j is the largest integer such that r ≥ j(m − 1) + n − 1 . If r ≥ i(m − 1) + n − 1 , then Combining (5), (6), (7) and (8) concludes the proof of (3).
The proof of (4) is similar. The main difference is that because ◻ approximately nm/2 terms while the only summation in (1) has approximately n/2 terms. Expressions (3) and (4) are nevertheless interesting because they allow us to derive an asymptotic result for S(m, n) when n is fixed and m → ∞ (see Theorem 3). This was not possible with (1). When n < 5 , expressions (3) and (4) reduce to particularly simple expressions.

Corollary 1
For n = 2, 3 or 4, the asymptotic behaviour of S(m, n) is easy to derive from this corollary, while the general case is covered by our next result.
Theorem 3 For all n ≥ 2 , when m → ∞ , S(m, n) is equal to m n−1 (n) + O(m n−2 ) where (n) is equal to when n is even, or to when n is odd.
Proof We first prove (9). Expression (3) for S(m, n) can also be written as For n fixed, (n) is the limit for n → ∞ of (11) divided by m n−1 , that is S(m, 4) = 2m 3 + m 3 .
are easy to compute for n between 2 and 100. Our implementation of (9) in Python takes approximately 0.029 seconds to compute (100) . For 100 < n < 1000 , computing (n) with our implementation of (n) takes longer but is still achievable (see Table 2). Computation of (n) for n larger than 1000 becomes problematic. Notice that, when n < 5 , Corollary 1 provides an asymptotic expression for S(m, n) that is tighter than that resulting from Theorem 3.