Ordered Bell numbers, Hermite polynomials, skew Young tableaux, and Borel orbits

  • Authors:

  • Mahir Bilen Can;Michael Joyce

  • Affiliations:

  • Tulane University, Department of Mathematics, New Orleans, LA, United States;Tulane University, Department of Mathematics, New Orleans, LA, United States

  • Venue:
  • Journal of Combinatorial Theory Series A
  • Year:
  • 2012

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Abstract

We give three interpretations of the number b of orbits of the Borel subgroup of upper triangular matrices on the variety X of complete quadrics. First, we show that b is equal to the number of standard Young tableaux on skew-diagrams. Then, we relate b to certain values of a modified Hermite polynomial. Third, we relate b to a certain cell decomposition on X previously studied by De Concini, Springer, and Strickland. Using these, we give asymptotic estimates for b as the dimension of the quadrics increases.