On xD-generalizations of stirling numbers and lah numbers via graphs and rooks

Sen Peng Eu, Yu Chang Liang, Tung Shan Fu, Tsai Lien Wong

研究成果: 雜誌貢獻期刊論文同行評審

3 引文 斯高帕斯(Scopus)

摘要

This paper studies the generalizations of the Stirling numbers of both kinds and the Lah numbers in association with the normal ordering problem in the Weyl algebra W = 〈x, D|Dx-xD = 1〉. Any word ω ∈ W with m x’s and n D’s can be expressed in the normally ordered form ω = xm−nΣk≥0{ω/k} x kDk , where {ω/k} is known as the Stirling number of the second kind for the word ω. This study considers the expansions of restricted words ω in W over the sequences {(xD)k}k≥0and {xDkxk−1}k≥0. Interestingly, the coefficients in individual expansions turn out to be generalizations of the Stirling numbers of the first kind and the Lah numbers. The coefficients will be determined through enumerations of some combinatorial structures linked to the words ω, involving decreasing forest decompositions of quasithreshold graphs and non-attacking rook placements on Ferrers boards. Extended to q-analogues, weighted refinements of the combinatorial interpretations are also investigated for words in the q-deformed Weyl algebra.

原文英語
期刊Electronic Journal of Combinatorics
24
發行號2
DOIs
出版狀態已發佈 - 2017 四月 13

ASJC Scopus subject areas

  • 理論電腦科學
  • 幾何和拓撲
  • 離散數學和組合
  • 計算機理論與數學
  • 應用數學

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