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<p>For a bounded and graded poset <i>P</i>, we show that if <i>P</i> is EL-shellable, then so is its <i>t</i>-fold Segre power <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(P^{(t)}=P\circ \cdots \circ P\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>P</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>=</mo> <mi>P</mi> <mo>∘</mo> <mo>⋯</mo> <mo>∘</mo> <mi>P</mi> </mrow> </math></EquationSource> </InlineEquation> (<i>t</i> factors), as defined by Björner and Welker (J. Pure Appl. Algebra, <b>198</b>(1-3), 43–55, 2005). Our EL-labeling leads to formulas for the rank-selected invariants of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(P^{(t)}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>P</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </msup> </math></EquationSource> </InlineEquation>, generalizing those given by Stanley for the subspace lattice (J. Combinatorial Theory Ser. A, <b>20</b>(3):336–356, 1976).</p>

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Segre Powers of Posets Preserve EL-Shellability

  • 期刊论文

  • 发表日期：

    2026年3月6日

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  • Yifei Li,
  • Sheila Sundaram

摘要

For a bounded and graded poset P, we show that if P is EL-shellable, then so is its t-fold Segre power \(P^{(t)}=P\circ \cdots \circ P\) P ( t ) = P ∘ ⋯ ∘ P (t factors), as defined by Björner and Welker (J. Pure Appl. Algebra, 198(1-3), 43–55, 2005). Our EL-labeling leads to formulas for the rank-selected invariants of \(P^{(t)}\) P ( t ) , generalizing those given by Stanley for the subspace lattice (J. Combinatorial Theory Ser. A, 20(3):336–356, 1976).