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Eulerian quasisymmetric functions and permutation statistics

Algebra & Discrete Mathematics

Speaker: John Shareshian, Washington University, St. Louis
Location: 1147 MSB
Start time: Fri, Feb 19 2010, 1:10PM

In joint work with Michelle Wachs, we defined Eulerian quasisymmetric functions Q_{n,j} as follows: Let w be a permutation in S_n, written in one line notation. If w has an excedance in position i, replace w(i) by a new symbol w(i)'. Call the new list of n symbols w'. Order the alphabet {1,2,...n,1',2',...,n'} by

Let DEX(w) be the set of all positions i such w'(i)>w'(i+1). Define F_w to be the quasisymmetric function obtained by adding all monomials m of degree n in the variables x_1,x_2,.... such that the indices of the variables appearing in m decrease weakly, with a strict decrease at any position lying in DEX(w). Finally, define Q_{n,j} to be the sum of F_w taken over all w in S_n with exactly j excedances.

It turns out that each Q_{n,j} is in fact a symmetric function. In fact, if we fix a conjugacy class C of S_n and restrict our sum to those w lying in C, this function remains symmetric. Using principal specialization (a standard tool for obtaining a power series in one variable from a symmetric function), we obtain from the Eulerian quasisymmetric functions various results about permutation statistics. I will discuss these results, along with some yet to be well understood coincidences of Eulerian quasisymmetric functions with other combinatorial objects.