Nonnegative Integer Matrices Research Articles

On a Transcendence Problem of K. Mahler

K. Mahler [8] has proposed the following problem. Let Ωr for r ≧ 1 be a sequence of n X n non-negative rational integer matrices. Each Ωr — (ωrij) defines a map Ωr : Cn —⟶ Cn by

The asymptotic number of non-negative integer matrices with given row and column sums

Asymptotics are obtained for the number of m× n non-negative integer matrices subject to the following constraints: (i) each row and each column sum is specified and bounded, (ii) the entries are bounded, and (iii) a specified “sparse” set of entries must be zero. The result can be interpreted in terms of incidence matrices for labeled digraphs.

Enumeration of plane partitions

Using some recent results involving Young tableaux and matrices of non-negative integers [10], it is possible to enumerate various classes of plane partitions by actual construction. One of the results is a simple proof of MacMahon's [12] generating function for plane partitions. Previous results of this type [12, 4, 3, 8, 7] involved complicated algebraic methods which did not reveal any intrinsic “reason” why the corresponding generating functions have such a simple form.

A theorem on cross-classification

(1)Cardinalities permitting, a finite set partitioned in two ways is the union of disjoint matrices all of whose rows and columns are systems of representatives of the first and second partition respectively. The proof uses: (2) A rectangular matrix of non-negative integers with a constant row sum and a constant column sum is the sum of matrices of non-negative integers for which the two constants are relatively prime.

Enumerating $(\bf{2+2})$-free posets by indistinguishable elements

A poset is said to be (2+2)-free if it does not contain an induced subposet that is isomorphic to 2+2, the union of two disjoint 2-element chains. Two elements in a poset are indistinguishable if they have the same strict up-set and the same strict down-set. Being indistinguishable defines an equivalence relation on the elements of the poset. We introduce the statistic maxindist, the maximum size of a set of indistinguishable elements. We show that, under a bijection of Bousquet-Melou et al., indistinguishable elements correspond to letters that belong to the same run in the so-called ascent sequence corresponding to the poset. We derive the generating function for the number of (2+2)-free posets with respect to both maxindist and the number of different strict down-sets of elements in the poset. Moreover, we show that (2+2)-free posets P with maxindist(P) at most k are in bijection with upper triangular matrices of nonnegative integers not exceeding k, where each row and each column contains a nonzero entry. (Here we consider isomorphic posets to be equal.) In particular, (2+2)-free posets P on n elements with maxindist(P)=1 correspond to upper triangular binary matrices where each row and column contains a nonzero entry, and whose entries sum to n. We derive a generating function counting such matrices, which confirms a conjecture of Jovovic, and we refine the generating function to count upper triangular matrices consisting of nonnegative integers not exceeding k and having a nonzero entry in each row and column. That refined generating function also enumerates (2+2)-free posets according to maxindist. Finally, we link our enumerative results to certain restricted permutations and matrices.