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. Let β n ( 0 , k ) and β n ( 1 , k ) denote the characters of Sn 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 the β n ( i , k ) The most elegant of these states that the generating function for the characters β 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 ɛnt$1. In the second part, we use these cycle index sum equations to examine the values of the characters β n ( 1 , 2 ) and β n ( 0 , 2 ) . We show that the values of β n ( 0 , 2 ) are simple multiples of the tangent numbers and that the restrictions of the β n ( 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 β n ( 1 , 2 ) a number of remarkable results arise. First it is shown that a series of polynomials {Pα(λ): σ € Sn} which are connected with our cycle index sum equations satisfy β n ( 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 ∏ n ( 1 , 2 ) is naturally modified to depend on σ then the polynomials pσ(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 of β n ( 0 , 2 ) to Sn−1 as a skew character. A consequence of this result is a simple combinatorial method for decomposing β n ( 0 , k ) into irreducibles.