44 research outputs found
Nonhomogeneous parking functions and noncrossing partitions
For each skew shape we define a nonhomogeneous symmetric function,
generalizing a construction of Pak and Postnikov. In two special cases, we show
that the coefficients of this function when expanded in the complete
homogeneous basis are given in terms of the (reduced) type of -divisible
noncrossing partitions. Our work extends Haiman's notion of a parking function
symmetric function.Comment: 11 pages, 3 figure
Three New Refined Arnold Families
The Springer numbers, introduced by Arnold, are generalizations of Euler
numbers in the sense of Coxeter groups. They appear as the row sums of a double
triangular array of integers, , defined recursively
by a boustrophedon algorithm. We say a sequence of combinatorial objects
is an Arnold family if is counted by . A
polynomial refinement of , together with the
combinatorial interpretations in several combinatorial structures was
introduced by Eu and Fu recently. In this paper, we provide three new Arnold
families of combinatorial objects, namely the cycle-up-down permutations, the
valley signed permutations and Knuth's flip equivalences on permutations. We
shall find corresponding statistics to realize the refined polynomial arrays