Partitions into Even and Odd Block Size and Some Unusual Characters of the Symmetric Groups

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