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Positivity Among P-partition Generating Functions

Abstract

We seek simple conditions on a pair of labeled posets that determine when the difference of their \((P,\omega )\)-partition enumerators is F-positive, i.e., positive in Gessel’s fundamental basis. This is a quasisymmetric analogue of the extensively studied problem of finding conditions on a pair of skew shapes that determine when the difference of their skew Schur functions is Schur-positive. We determine necessary conditions and separate sufficient conditions for F-positivity, and show that a broad operation for combining posets preserves positivity properties. We conclude with classes of posets for which we have conditions that are both necessary and sufficient.

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Acknowledgements

We are grateful to Christophe Reutenauer for asking the second author about equality of \((P,\omega )\)-partition enumerators, of which this project is an outgrowth. We thank the anonymous referee for helpful comments, including a reminder of the connection to the 0-Hecke algebra. This paper is based on the first author’s undergraduate honors thesis at Bucknell University where his research was funded by the Hoover Math Scholarship and the Department of Mathematics. Portions of this paper were written while the second author was on sabbatical at Université de Bordeaux; he thanks LaBRI for its hospitality. Computations were performed using SageMath [40].

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Correspondence to Peter R. W. McNamara.

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Communicated by Vasu Tewari

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Lesnevich, N.R.T., McNamara, P.R.W. Positivity Among P-partition Generating Functions. Ann. Comb. 26, 171–204 (2022). https://doi.org/10.1007/s00026-021-00563-2

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Keywords

  • P-partition
  • Labeled poset
  • Quasisymmetric function
  • F-positive
  • Linear extension