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Equality of P-Partition Generating Functions

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Abstract

To every labeled poset (P, ω), one can associate a quasisymmetric generating function for its (P, ω)-partitions. We ask: when do two labeled posets have the same generating function? Since the special case corresponding to skew Schur function equality is still open, a complete classification of equality among (P, ω) generating functions is likely too much to expect. Instead, we determine necessary conditions and separate sufficient conditions for two labeled posets to have equal generating functions. We conclude with a classification of all equalities for labeled posets with small numbers of linear extensions.

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References

  1. Barekat, F., Reiner, V., van Willigenburg, S.: Corrigendum to “Coincidences among skew Schur functions” [Adv. Math. 216 (1) (2007) 118–152]. Adv. Math. 220(5), 1655–1656 (2009)

  2. Billera L.J., Thomas H., van Willigenburg S.: Decomposable compositions, symmetric quasisymmetric functions and equality of ribbon Schur functions. Adv.Math. 204(1), 204–240 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  3. Ehrenborg R.: On posets and Hopf algebras. Adv. Math. 119(1), 1–25 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  4. Gessel, I.M.: Multipartite P-partitions and inner products of skew Schur functions. In: Greene, C. (ed.) Combinatorics and Algebra. Contemp. Math. Vol. 34, pp. 289–301. Amer. Math. Soc. Providence, RI (1984)

  5. Gessel, I.M.: Quasi-symmetric functions. Unpublished manuscript (1990)

  6. Gutschwager C.: Equality of multiplicity free skew characters. J. Algebraic Combin. 30(2), 215–232 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  7. Lam T., Pylyavskyy P.: Cell transfer and monomial positivity. J. Algebraic Combin. 26(2), 209–224 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  8. Lam T., Pylyavskyy P.: P-partition products and fundamental quasi-symmetric function positivity. Adv. Appl. Math. 40(3), 271–294 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  9. Malvenuto, C.: Produits et coproduits des fonctions quasi-symétriques et de l’algèbre des descentes. Ph.D. thesis. Laboratoire de Combinatoire et d’Informatique Mathématique, Université du Québec à Montréal (1993)

  10. Malvenuto C., Reutenauer C.: Duality between quasi-symmetric functions and the Solomon descent algebra. J. Algebra 177(3), 967–982 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  11. Malvenuto C., Reutenauer C.: Plethysm and conjugation of quasi-symmetric functions. Discrete Math. 193(1-3), 225–233 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  12. McNamara P.: Cylindric skew Schur functions. Adv. Math. 205(1), 275–312 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  13. McNamara P.R.W., van Willigenburg S.: Towards a combinatorial classification of skew Schur functions. Trans. Amer. Math. Soc. 361(8), 4437–4470 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  14. Reiner V., Shaw K.M., van Willigenburg S.: Coincidences among skew Schur functions. Adv. Math. 216(1), 118–152 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  15. Stanley, R.P.: Ordered structures and partitions. Ph.D. thesis, Harvard University, Cambridge (1971)

  16. Stanley, R.P.: Ordered Structures and Partitions. Memoirs of the American Mathematical Society, No. 119. American Mathematical Society, Providence, R.I. (1972)

  17. Stanley, R.P.: Enumerative Combinatorics. Vol. I. Corrected reprint of the 1986 original. Cambridge University Press, Cambridge/New York (1997)

  18. Stanley, R.P.: Enumerative Combinatorics. Vol. 2. Cambridge Stud. Adv. Math., Vol. 62. Cambridge University Press, Cambridge (1999)

  19. Stanley, R.P.: Enumerative Combinatorics. Vol. 1. Cambridge Stud. Adv. Math., Vol. 49. Second Edition. Cambridge University Press, Cambridge (2012)

  20. Stein, W.A. et al.: Sage Mathematics Software (version 4.5.2). The Sage Development Team. http://www.sagemath.org (2010)

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Authors and Affiliations

  1. Department of Mathematics, Bucknell University, Lewisburg, PA, 17837, USA

    Peter R. W. McNamara

  2. Department of Mathematics, The Pennsylvania State University, University Park, PA, 16802, USA

    Ryan E. Ward

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  1. Peter R. W. McNamara
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  2. Ryan E. Ward
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Correspondence to Peter R. W. McNamara.

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McNamara, P.R.W., Ward, R.E. Equality of P-Partition Generating Functions. Ann. Comb. 18, 489–514 (2014). https://doi.org/10.1007/s00026-014-0236-7

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  • DOI: https://doi.org/10.1007/s00026-014-0236-7

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