Eulerian numbers, Newcomb's problem and representations of symmetric groups

Abstract

Consider a set of n positive integers consisting of μ 1 1's, μ 2 2's,…, μ r r's. If the integer in the ith place in an arrangement σ of this set is σ( i), and a non-rise in σ is defined as σ( i+1)⩽ σ( i), a problem that suggests itself is the determination of the number of arrangements σ with k non-rises. When each μ i is unity, the problem is that of finding the number A( n, k) of permutations of distinct integers 1, 2,…, n with k descents, a descent being defined as σ( i+1)< σ( i). The number A( n, k) is known as an Eulerian number. The problem of finding the number of arrangements with k non-rises of the more general set, when not all of μ i are unity, has appeared in the literature as one part of a problem on dealing a pack of cards, this having been proposed by the American astronomer Simon Newcomb (1835–1909). Both the Eulerian numbers and Newcomb's problem have accumulated a substantial literature. The present paper considers these topics from an entirely new stand-point, that of representations of the symmetric group. This approach yields a well-known recurrence for the Eulerian numbers and a known formula for them in terms of Stirling numbers. It also gives the solutions of the Newcomb problem and some recurrences between these solutions, not all of which have been found earlier. A simple connection is found between Stirling numbers and the Kostka numbers of symmetric group representation theory. The Eulerian numbers can also be expressed in terms of the Kostka numbers. The idea which is novel in this treatment and recurs almost as a motif throughout the paper is that of a skew-hook. This occurs in the first place in a very natural way as a picture of the rises and non-rises of σ, with the nodes of the skew-hook labelled successively as σ(1), σ(2),…. As the paper develops, a new form of skew-hook associated with σ emerges. This does not in general depict the rises and non-rises of σ, and it is now the edges, not the nodes, which carry integer labels. A new type of combinatorial number, here called a ψ-function, arises from these edge-labelled skew-hooks. The ψ-functions are intimately related to the Eulerian numbers and the Newcomb solutions and may have further combinatorial applications. The skew-hook treatment casts fresh light on MacMahon's solution of the Newcomb problem and on his “new symmetric functions”, and, if σ( i)− σ( i+1)⩾ s defines an s-descent in σ, on the enumeration of permutations with k s-descents. Also some characters of the symmetric group with interesting properties and recurrences arise in the course of the paper.