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Descent polynomials

Bruce Sagan, Michigan State University
Thursday, November 2, 2017 - 10:30am
PDL C-36

A permutation \$\pi=\pi_1\dots\pi_n\$ in the symmetric group \${\mathfrak S}_n\$ has descent set 
\${\rm Des} \,\,\pi=\{i\ |\ \pi_i>\pi_{i+1}\}\$. Given a set \$I\$ of positive integers and \$n>\max I\$, the descent polynomial of \$I\$ is the cardinality \$d(I;n)=\#\{\pi\in{\mathfrak S}_n\ |\ {\rm Des}\,\,\pi=I\}\$.
In 1915, MacMahon proved, using the Principle of Inclusion and Exclusion, that this is a polynomial in \$n\$. Amazingly, since then properties of this polynomial do not seem to have been studied at all in the literature. We will investigate the descent polynomial in terms of its degree, coefficients when expanded in a basis of binomial coefficients, and roots. We will also mention a connection with the peak polynomial of Billey, Burdzy, and Sagan. This is joint work with Alexander Diaz-Lopez, Pamela Harris, Erik Insko, and Mohamed Omar.

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