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My paper "Alternating permutations and symmetric functions" at http://math.mit.edu/~rstan/papers/altenum.pdf enumerates certain classes of alternating permutations, such as those whose inverse is alternating or which have a given cycle type. The key result on which the proofs depend is that for a certain (reducible) character $\chi$ of the symmetric group $S_n$ the character values are all either 0 or $\pm E_k$, where $E_k$ is an Euler number (the number of alternating permutations of $1,2,\dots,k$). This character was first considered by Foulkes. (For the cognoscenti, it is the character corresponding to a ribbon staircase.) I am interested in whether there are $q$-analogues of my results, e.g., enumerating the various classes of permutations by some statistic such as major index or number of inversions. A natural approach would be to find a $q$-analogue of Foulkes' character $\chi$. Some first attempts such as the Hall-Littlewood or Hecke algebra analogues of the characters of $S_n$ don't seem to work.

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  • $\begingroup$ There is a way to define q-Bernoulli numbers using a q-analog of the usual Bernoulli umbra (see the works of Carlitz on q-Bernoulli numbers). Maybe something similar could work for the Euler umbra you are using ? $\endgroup$
    – F. C.
    Feb 15, 2014 at 21:10

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