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File Placements, Fractional Matchings, and Normal Ordering


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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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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Schork, M. File Placements, Fractional Matchings, and Normal Ordering. Ann. Comb. (2022).

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

Mathematics Subject Classification

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