# Partitions into even and odd block size and some unusual characters of the symmetric groups

Journal Article (Journal Article)

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

### Cited Authors

• Calderbank, AR; Robinson, RW; Hanlon, P

### Published Date

• January 1, 1986

• s3-53 / 2

• 288 - 320

• 1460-244X

• 0024-6115

### Digital Object Identifier (DOI)

• 10.1112/plms/s3-53.2.288

• Scopus