enumerating (2+2)-free posets by the number of minimal elements and other statistics sergey kitaev...
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Enumerating (2+2)-free posets by the number of minimal elements and other statistics
Sergey KitaevReykjavik University
Joint work with
Jeff RemmelUniversity of California, San Diego
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
Ascent sequencesNumber 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.
Mireille Bousquet-Mélou
Anders Claesson
Mark Dukes
SK
Unlabeled (2+2)-free posets, ascent sequences, and pattern avoiding permutations
Mireille Bousquet-Mélou
Anders Claesson
Mark Dukes
SK
Unlabeled (2+2)-free posets, ascent sequences, and pattern avoiding permutations
Robert Parviainen
Mireille Bousquet-MélouAnders 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
Mireille Bousquet-MélouAnders Claesson
Mark Dukes
SK
Unlabeled (2+2)-free posets, ascent sequences, and pattern avoiding permutations
Jeff Remmel
The present talk
Mireille Bousquet-MélouAnders 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
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
_ _
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.
Unlabeled (2+2)-free posets
How can one decompose a (2+2)-free poset?
Unlabeled (2+2)-free posets
2
Unlabeled (2+2)-free posets
1 1 3
1 0 1
Read labels backwards: (0, 1, 0, 1, 3, 1, 1, 2) – an ascent sequence!
Removing last point gives one extra 0.
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)
Some statistics preserved under the bijection
(0, 1, 0, 1, 3, 1, 1, 2)
(0, 1, 0, 1, 3, 1, 1, 2 )
(0, 1, 0, 1, 3, 1, 1, 2)
min zeros
min maxlevel
last element
(0, 3, 0, 1, 4, 1, 1, 2)
Level distri-bution
letter distributionin modif. sequence
Some statistics preserved under the bijection
(0, 1, 0, 1, 3, 1, 1, 2)
(0, 1, 0, 1, 3, 1, 1, 2)
(0, 1, 0, 1, 3, 1, 1, 2)
highestlevel
number of ascents
(0, 3, 0, 1, 4, 1, 1, 2)
right-to-left maxin mod. sequencemax
compo-nents
Components inmodif. sequence
(0, 3, 0, 1, 4, 1, 1, 2)
A generalization of the generating function
lds=size of last non-trivial downset
...
minmaxmin
The main result in this talk (SK & J. Remmel, 2009):
The corresponding posets:
A conjecture (SK & J. Remmel, 2009):
Compare to
Posets avoiding and
Ascent sequences are restricted as follows:
m-1, where m is the max element here
Catalan many
Catalan many
Hilmar HaukurGuðmundsson
Posets avoiding and
Self modified ascent sequences
Bayoumi, El-Zahar, Khamis (1989)
Thank you for your attention!