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

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) = Such posets arise as interval orders (Fishburn): P. C. Fishburn, Intransitive indifference with unequal indifference intervals, J. Math. Psych. 7 (1970) 144–149. bad guy good guy

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

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

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

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

8. Mireille Bousquet-Mélou Anders Claesson Mark Dukes SK Unlabeled (2+2)-free posets, ascent sequences, and pattern avoiding permutations 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. 113 101 Read labels backwards: (0, 1, 0, 1, 3, 1, 1, 2) – an ascent sequence! Removing last point gives one extra 0.

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 (0, 1, 0, 1, 3, 1, 1, 2) min zeros min max level last element (0, 3, 0, 1, 4, 1, 1, 2) Level distri- bution letter distribution in modif. sequence

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

18. A generalization of the generating function lds=size of last non-trivial downset... minmaxmin

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

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

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

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

23. Thank you for your attention!