Partitions into even and odd block size and some unusual characters of the symmetric groups
For each n and k, let Π(i, k) denote the poset of all partitions of n having every block size congruent to i mod k. Attach to Πn(i, k) a unique maximal or minimal element if it does not already have one, and denote the resulting poset Πn(i, k). Results of Björner, Sagan, and Wachs show that Πn(0, k) and Πn(1, k) are lexicographically shellable, and hence Cohen-Macaulay. Let βn(0, k) and βn(0, k) denote the characters of S„ acting on the unique non-vanishing reduced homology groups of Πn(0, k) and Πn(1, k).This paper is divided into three parts. In the first part, we use combinatorial methods to derive defining equations for the generating functions of the character values of the βn(i, k). The most elegant of these states that the generating function for the characters βn(1, k) (t = 0, 1,…) is the inverse in the composition ring (or plethysm ring) to the generating function for the corresponding trivial characters εni+l. In the second part, we use these cycle index sum equations to examine the values of the characters βn(1, 2) and βn(0, 2). We show that the values of βn(0, 2) are simple multiples of the tangent numbers and that the restrictions of the βn(0, 2) to Sn-1 are the skew characters examined by Foulkes (whose values are always plus or minus a tangent number). In the case βn(0, 2) a number of remarkable results arise. First it is shown that a series of polynomials (pσ(λ): σeSn) which are connected with our cycle index sum equations satisfy βn(1, 2)(σ) = pσ (0) or pσ (1) depending on whether n is odd or even. Next it is shown that the pσ(λ) have integer roots which obey a simple recursion. Lastly it is shown that the pσ(λ)have a combinatorial interpretation. If the rank function of Πn(1, k) is naturally modified to depend on a then the polynomials pσ(λ) are the Birkhoff polynomials of the fixed point posets Πn(1, k)σ In the last part we prove a conjecture of R. P. Stanley which indentifies the restriction of βn(0, 2) to Sn-1, as a skew character. A consequence of this result is a simple combinatorial method for decomposing βn(0, k) into irreducibles. © 1986 Oxford University Press.
Calderbank, AR; Robinson, RW; Hanlon, P
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