Abstract
Bóna conjectured that the descent polynomials on $(n-2)$-stack sortable permutations have only real zeros. Brändén proved this conjecture by establishing a more general result. In this paper, we give another proof of Brändén's result by using the theory of $s$-Eulerian polynomials recently developed by Savage and Visontai.
Highlights
Suppose that w = w1 · · · wn is a permutation of a set of distinct numbers and wi is the maximal number of {w1, . . . , wn}
Bona conjectured that the descent polynomials on (n − 2)-stack sortable permutations have only real zeros
By Lemma 2.1, we see that the polynomial sequence {fi(x)}mi=1 is pairwise interlacing
Summary
Bona conjectured that the descent polynomials on (n − 2)-stack sortable permutations have only real zeros. We give another proof of Branden’s result by using the theory of s-Eulerian polynomials recently developed by Savage and Visontai. The descent polynomial Wn,t(x) has only real zeros for any integer 1 t n − 1. Kn(x) = An(x) + kx An−2(x) has only real zeros.
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