Journal of Integer Sequences, Vol. 25 (2022), Article 22.1.1

Cyclic and Linear Graph Partitions and Normal Ordering

Ken Joffaniel Gonzales
Department of Physical Sciences and Mathematics
University of the Philippines Manila
Manila 1000


The Stirling number of a simple graph is the number of partitions of its vertex set into a specific number of non-empty independent sets. Engbers et al. showed that the coefficients in the normal ordering of a word w in the alphabet {x,D} subject to the relation Dx = xD + 1 are equal to the Stirling number of certain graphs constructed from w. In this paper, we introduce graphical versions of the Stirling numbers of the first kind and the Lah numbers and show how they occur as coefficients in other normal ordering settings. We also obtain identities involving their q-analogues.

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(Concerned with sequences A111596 A132393.)

Received June 15 2021; revised version received November 1 2021. Published in Journal of Integer Sequences, December 28 2021.

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