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!