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Enumerating (2+2)-free posets by the number of minimal elements and other statistics

Enumerating (2+2)-free posets by the number of minimal elements and other statistics. Joint work with. Jeff Remmel University of California, San Diego. Sergey Kitaev Reykjavik University. Unlabeled (2+2)-free posets. A partially ordered set is called (2+2)-free if it contains no induced

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Enumerating (2+2)-free posets by the number of minimal elements and other statistics

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  1. Enumerating (2+2)-free posets by the number of minimal elements and other statistics Joint work with Jeff Remmel University of California, San Diego Sergey Kitaev Reykjavik University

  2. Unlabeled (2+2)-free posets A partially ordered set is called (2+2)-free if it contains no induced sub-posets isomorphic to (2+2) = bad guy good guy Such posets arise as interval orders (Fishburn): P. C. Fishburn, Intransitive indifference with unequal indifference intervals, J. Math. Psych. 7 (1970) 144–149.

  3. Ascent sequences Number of ascents in a word: asc(0, 0, 2, 1, 1, 0, 3, 1, 2, 3) = 4 (0,0,2,1,1,0,3,1,2,3) is not an ascent sequence, whereas (0,0,1,0,1,3,0) is.

  4. Unlabeled (2+2)-free posets, ascent sequences, and pattern avoiding permutations Mireille Bousquet-Mélou Mark Dukes SK Anders Claesson

  5. Unlabeled (2+2)-free posets, ascent sequences, and pattern avoiding permutations Mireille Bousquet-Mélou Mark Dukes SK Anders Claesson Robert Parviainen

  6. Unlabeled (2+2)-free posets, ascent sequences, and pattern avoiding permutations Mireille Bousquet-Mélou Mark Dukes Anders Claesson SK Invited talk at the AMS-MAA joint mathematics meeting Svante Linusson

  7. Unlabeled (2+2)-free posets, ascent sequences, and pattern avoiding permutations Mireille Bousquet-Mélou Mark Dukes Anders Claesson SK The present talk Jeff Remmel

  8. Unlabeled (2+2)-free posets, ascent sequences, and pattern avoiding permutations Mireille Bousquet-Mélou Mark Dukes Anders Claesson SK A direct encoding of Stoimenow’s matchings as ascent sequences

  9. Overview of results by Bousquet-Mélou et al. (2008) Bijections (respecting several statistics) between the following objects unlabeled (2+2)-free posets on n elements pattern-avoiding permutations of length n ascent sequences of length n linearized chord diagrams with n chords = certain involutions Closed form for the generating function for these classes of objects _ _ Pudwell’s conjecture (on permutations avoiding 31524) is settled using modified ascent sequences

  10. Unlabeled (2+2)-free posets Theorem. (easy to prove) A poset is (2+2)-free iff the collection of strict down-sets may be linearly ordered by inclusion.

  11. Unlabeled (2+2)-free posets How can one decompose a (2+2)-free poset?

  12. Unlabeled (2+2)-free posets 2

  13. Unlabeled (2+2)-free posets 1 1 3 1 0 1 Removing last point gives one extra 0. Read labels backwards: (0, 1, 0, 1, 3, 1, 1, 2) – an ascent sequence!

  14. Theorem. There is a 1-1 correspondence between unlabeled (2+2)-free posets on n elements and ascent sequences of length n. (0, 1, 0, 1, 3, 1, 1, 2)

  15. Some statistics preserved under the bijection min zeros (0, 1, 0, 1, 3, 1, 1, 2) min max level last element (0, 1, 0, 1, 3, 1, 1, 2 ) Level distri- bution letter distribution in modif. sequence (0, 1, 0, 1, 3, 1, 1, 2) (0, 3, 0, 1,4, 1, 1, 2)

  16. Some statistics preserved under the bijection highest level number of ascents (0, 1, 0, 1, 3, 1, 1, 2) right-to-left max in mod. sequence max (0, 1, 0, 1, 3, 1, 1, 2) (0, 3, 0, 1, 4, 1, 1, 2) compo- nents Components in modif. sequence (0, 1, 0, 1, 3, 1, 1, 2) (0, 3, 0, 1, 4, 1, 1, 2)

  17. A generalization of the generating function ... min minmax lds=size of last non-trivial downset

  18. The main result in this talk (SK & J. Remmel, 2009): The corresponding posets:

  19. A conjecture (SK & J. Remmel, 2009): Compare to

  20. Posets avoiding and Catalan many Ascent sequences are restricted as follows: m-1, where m is the max element here Catalan many Hilmar Haukur Guðmundsson

  21. Posets avoiding and Bayoumi, El-Zahar, Khamis (1989) Self modified ascent sequences

  22. Thank you for your attention!

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