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File Placements, Fractional Matchings, and Normal Ordering

Abstract

In this paper, the bijection between k-rook placements on a Ferrers board and k-matchings in the associated bipartite graph is extended to a bijection between k-file placements and certain fractional matchings. Using the latter bijection, a new interpretation is given for the normal ordering coefficients in the shift algebra. Several further results concerning normal ordering in the shift algebra are derived.

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Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.

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Acknowledgements

The author wishes to express his grateful thanks to the anonymous referees for their comments and suggestions helping to improve this paper. In particular, one of the referees pointed out identity (3.14) and its concrete realization mentioned in Remark 3.12.

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Correspondence to Matthias Schork.

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Communicated by Bridget Tenner.

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Schork, M. File Placements, Fractional Matchings, and Normal Ordering. Ann. Comb. (2022). https://doi.org/10.1007/s00026-022-00599-y

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Keywords

  • Ferrers boards
  • Rook placements
  • File placements
  • Weyl algebra
  • Shift algebra
  • Normal ordering
  • Generalized Stirling numbers

Mathematics Subject Classification

  • 05A10
  • 05A19
  • 05C70
  • 05C72
  • 11B73
  • 16S99