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<p>In an earlier paper, we defined and studied <i>q</i>-analogs of the Stirling numbers of both types for the Coxeter group of type <i>B</i>. In the present work, we show how this approach can be extended to all irreducible complex reflection groups <i>G</i>. The Stirling numbers of the first and second kind are defined via the Whitney numbers of the first and second kind, respectively, of the intersection lattice of <i>G</i>. For the groups <i>G</i>(<i>m</i>,&#xa0;<i>p</i>,&#xa0;<i>n</i>), these numbers and polynomials can be given combinatorial interpretations in terms of various statistics. The ordered version of the <i>q</i>-Stirling numbers of the second kind also show up in conjectured Hilbert series for certain super coinvariant algebras.</p>

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Stirling Numbers for Complex Reflection Groups

  • 期刊论文

  • 发表日期：

    2025年3月24日

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  • Bruce E. Sagan,
  • Joshua P. Swanson

摘要

In an earlier paper, we defined and studied q-analogs of the Stirling numbers of both types for the Coxeter group of type B. In the present work, we show how this approach can be extended to all irreducible complex reflection groups G. The Stirling numbers of the first and second kind are defined via the Whitney numbers of the first and second kind, respectively, of the intersection lattice of G. For the groups G(m, p, n), these numbers and polynomials can be given combinatorial interpretations in terms of various statistics. The ordered version of the q-Stirling numbers of the second kind also show up in conjectured Hilbert series for certain super coinvariant algebras.