2 research outputs found
Some determinants of path generating functions
We evaluate four families of determinants of matrices, where the entries are
sums or differences of generating functions for paths consisting of up-steps,
down-steps and level steps. By specialisation, these determinant evaluations
have numerous corollaries. In particular, they cover numerous determinant
evaluations of combinatorial numbers - most notably of Catalan, ballot, and of
Motzkin numbers - that appeared previously in the literature.Comment: 35 pages, AmS-TeX; minor corrections; final version to appear in Adv.
Appl. Mat
Eulerian quasisymmetric functions and cyclic sieving
It is shown that a refined version of a q-analogue of the Eulerian numbers
together with the action, by conjugation, of the subgroup of the symmetric
group generated by the -cycle on the set of permutations
of fixed cycle type and fixed number of excedances provides an instance of the
cyclic sieving phenonmenon of Reiner, Stanton and White. The main tool is a
class of symmetric functions recently introduced in work of two of the authors.Comment: 30 page