Reference no: EM131085714
Math 121c: Topics in Geometric Combinatorics, Spring 2012 Problems-
For π ∈ Sn, the descent set of π is
Des(π) := {j ∈ [n - 1] : π(j) > π(j + 1)}.
The descent statistic is des(π) = |Des(π)|. The descent statistic is encoded in the Eulerian polynomial ∑π∈Sn tdes(π), and the most basic identity for Eulerian polynomials is
∑k≥0(k + 1)ntk = ∑π∈S_n tdes(π)/(1 - t)n+1.
Prove this by proving the following more general result:
∑k≥0j=1∏n[k + 1]zjz0k = ∑π∈S_n(∏j∈Des(π)z0zπ(1)zπ(2)· · · zπ(j)/j=0∏n(1 - z0zπ(1)zπ(2)· · · zπ(j))),
where [m]q = 1 + q + q2 + · · · + qm-1.
(Hint: Integer point transform on the cone over [0, 1]n, together with triangulation.)
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