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Staircase skew Schur functions are Schur \(P\)-positive. (English) Zbl 1253.05137

Summary: We prove Stanley’s conjecture that, if \(\delta_n\) is the staircase shape, then the skew Schur functions \(s_{{\delta_n}/\mu}\) are non-negative sums of Schur \(P\)-functions. We prove that the coefficients in this sum count certain fillings of shifted shapes. In particular, for the skew Schur function \(s_{{\delta_n}/ \delta_{n-2}}\), we discuss connections with Eulerian numbers and alternating permutations.

MSC:

05E05 Symmetric functions and generalizations
05A05 Permutations, words, matrices
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