Ascent sequences avoiding pairs of patterns Lara Pudwell Introduction & History Pairs of Length 3 Patterns Unbalanced equivalences An Erd˝ os-Szekeres-like Theorem Other sequences Dyck paths Generating trees Onward... Ascent sequences avoiding pairs of patterns Lara Pudwell faculty.valpo.edu/lpudwell joint work with Andrew Baxter Permutation Patterns 2014 East Tennessee State University July 7, 2014
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Ascent sequencesavoiding pairs of
patterns
Lara Pudwell
Introduction &History
Pairs of Length 3PatternsUnbalanced equivalences
An Erdos-Szekeres-likeTheorem
Other sequences
Dyck paths
Generating trees
Onward...
Ascent sequences avoiding pairs ofpatterns
Lara Pudwellfaculty.valpo.edu/lpudwell
joint work with
Andrew Baxter
Permutation Patterns 2014East Tennessee State University
DefinitionAn ascent in the string x1 · · · xn is a position i such thatxi < xi+1.
Example:
01024 01024 01024
Definitionasc(x1 · · · xn) is the number of ascents of x1 · · · xn.
Example: asc(01024) = 3
Ascent sequencesavoiding pairs of
patterns
Lara Pudwell
Introduction &History
Pairs of Length 3PatternsUnbalanced equivalences
An Erdos-Szekeres-likeTheorem
Other sequences
Dyck paths
Generating trees
Onward...
Ascents
DefinitionAn ascent in the string x1 · · · xn is a position i such thatxi < xi+1.
Example:
01024 01024 01024
Definitionasc(x1 · · · xn) is the number of ascents of x1 · · · xn.
Example: asc(01024) = 3
Ascent sequencesavoiding pairs of
patterns
Lara Pudwell
Introduction &History
Pairs of Length 3PatternsUnbalanced equivalences
An Erdos-Szekeres-likeTheorem
Other sequences
Dyck paths
Generating trees
Onward...
Ascent SequencesDefinitionAn ascent sequence is a string x1 · · · xn of non-negativeintegers such that:I x1 = 0I xn ≤ 1 + asc(x1 · · · xn−1) for n ≥ 2An is the set of ascent sequences of length n
Ascent SequencesDefinitionAn ascent sequence is a string x1 · · · xn of non-negativeintegers such that:I x1 = 0I xn ≤ 1 + asc(x1 · · · xn−1) for n ≥ 2An is the set of ascent sequences of length n
= 78 pairsI at least 35 different sequences aσ,τ (n)
16 sequences in OEISI 3 sequences from Duncan/SteingrımssonI 1 eventually zeroI 1 from pattern-avoiding set partitionsI 3 from pattern-avoiding permutationsI 1 sequence from Mansour/Shattuck
Proof scribble:generating tree → recurrence → system of functionalequations → experimental solution → plug in for catalyticvariables
Conjecture (Duncan & Steingrımsson)
a0021(n) = a1012(n) =n−1∑k=0
(n−1k)Ck
Note: Proving this would complete Wilf classification of 4patterns.
Ascent sequencesavoiding pairs of
patterns
Lara Pudwell
Introduction &History
Pairs of Length 3PatternsUnbalanced equivalences
An Erdos-Szekeres-likeTheorem
Other sequences
Dyck paths
Generating trees
Onward...
Avoiding 201 and 210
Theorem
a201,210(n) =n−1∑k=0
(n−1k)Ck
Proof scribble:generating tree → recurrence → system of functionalequations → experimental solution → plug in for catalyticvariablesConjecture (Duncan & Steingrımsson)
a0021(n) = a1012(n) =n−1∑k=0
(n−1k)Ck
Note: Proving this would complete Wilf classification of 4patterns.
Ascent sequencesavoiding pairs of
patterns
Lara Pudwell
Introduction &History
Pairs of Length 3PatternsUnbalanced equivalences
An Erdos-Szekeres-likeTheorem
Other sequences
Dyck paths
Generating trees
Onward...
A familiar sequence...
Conjecture (Duncan & Steingrımsson)
a0021(n) = a1012(n) =n−1∑k=0
(n−1k)Ck
Theorem (Mansour & Shattuck)
a1012(n) =n−1∑k=0
(n−1k)Ck
Theorem
a0021(n) =n−1∑k=0
(n−1k)Ck
Proof: Similar technique to a201,210(n).
Ascent sequencesavoiding pairs of
patterns
Lara Pudwell
Introduction &History
Pairs of Length 3PatternsUnbalanced equivalences
An Erdos-Szekeres-likeTheorem
Other sequences
Dyck paths
Generating trees
Onward...
A familiar sequence...
Conjecture (Duncan & Steingrımsson)
a0021(n) = a1012(n) =n−1∑k=0
(n−1k)Ck
Theorem (Mansour & Shattuck)
a1012(n) =n−1∑k=0
(n−1k)Ck
Theorem
a0021(n) =n−1∑k=0
(n−1k)Ck
Proof: Similar technique to a201,210(n).
Ascent sequencesavoiding pairs of
patterns
Lara Pudwell
Introduction &History
Pairs of Length 3PatternsUnbalanced equivalences
An Erdos-Szekeres-likeTheorem
Other sequences
Dyck paths
Generating trees
Onward...
Summary and Future work
I 16 pairs of 3-patterns appear in OEIS.I Erdos-Szekeres analog for ascent sequences.I New bijective proof connecting 100,101-avoiders to
Dyck paths.I Completed Wilf classification of 4-patterns.I Open:
I 19 sequences from pairs of 3-patterns not in OEIS.I Bijective explanation that a021,102(n) = |Sn(123, 3241)|.
Forthcoming:I Enumeration schemes for pattern-avoiding ascent
sequencesI Details on a201,210(n) and a0021(n)I More bijections with other combinatorial objects?
Ascent sequencesavoiding pairs of
patterns
Lara Pudwell
Introduction &History
Pairs of Length 3PatternsUnbalanced equivalences
An Erdos-Szekeres-likeTheorem
Other sequences
Dyck paths
Generating trees
Onward...
Summary and Future work
I 16 pairs of 3-patterns appear in OEIS.I Erdos-Szekeres analog for ascent sequences.I New bijective proof connecting 100,101-avoiders to
Dyck paths.I Completed Wilf classification of 4-patterns.I Open:
I 19 sequences from pairs of 3-patterns not in OEIS.I Bijective explanation that a021,102(n) = |Sn(123, 3241)|.
Forthcoming:I Enumeration schemes for pattern-avoiding ascent
sequencesI Details on a201,210(n) and a0021(n)I More bijections with other combinatorial objects?
Ascent sequencesavoiding pairs of
patterns
Lara Pudwell
Introduction &History
Pairs of Length 3PatternsUnbalanced equivalences
An Erdos-Szekeres-likeTheorem
Other sequences
Dyck paths
Generating trees
Onward...
ReferencesI A. Baxter and L. Pudwell, Ascent sequences avoiding pairs of patterns,
arXiv:1406.4100, submitted.I M. Bousquet-Melou, A. Claesson, M. Dukes, S. Kitaev, (2+2)-free
posets, ascent sequences, and pattern avoiding permutations, J.Combin. Theory Ser. A 117 (2010), 884–909.
I D. Callan, T. Mansour, and M. Shattuck. Restricted ascent sequencesand Catalan numbers. arXiv:1403.6933, March 2014.
I P. Duncan and E. Steingrımsson. Pattern avoidance in ascentsequences. Electronic J. Combin. 18(1) (2011), #P226 (17pp).
I T. Mansour and M. Shattuck. Restricted partitions and generalizedCatalan numbers. Pure Math. Appl. (PU.M.A.) 22 (2011), no. 2,239–251. 05A18 (05A15)
I T. Mansour and M. Shattuck. Some enumerative results related toascent sequences. Discrete Mathematics 315-316 (2014), 29–41.
I V. Vatter. Finitely labeled generating trees and restricted permutations,J. Symb. Comput. 41 (2006), 559–572.