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Enumeration schemes for dashed Enumeration schemes for permutation patterns dashed permutation patterns Lara Pudwell Dashed Patterns Definition Andrew Baxter Brief History Rutgers University Enumeration Schemes Definition &

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Enumeration schemes for dashed permutation patterns Lara Pudwell Dashed Patterns

Definition Brief History

Enumeration Schemes

Definition & Notation Example Algorithm Usefulness

Results

Guaranteed Schemes Success Rates Sequences

Enumeration schemes for dashed permutation patterns

Andrew Baxter Rutgers University Lara Pudwell∗ Valparaiso University Permutation Patterns 2011 June 21, 2011

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Enumeration schemes for dashed permutation patterns Lara Pudwell Dashed Patterns

Definition Brief History

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Definition & Notation Example Algorithm Usefulness

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Dashed Patterns Dashed Pattern A dashed permutation pattern is a permutation π ∈ Sn where each pair of consecutive numbers may or may not have a dash between them. (also called generalized or vincular patterns) No dash – numbers must appear adjacently Dash – numbers may appear arbitrarily far apart. Example 251643 contains 1 − 2 − 3 and 12 − 3 but avoids 1 − 23.

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Enumeration schemes for dashed permutation patterns Lara Pudwell Dashed Patterns

Definition Brief History

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Definition & Notation Example Algorithm Usefulness

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Dashed Patterns – Brief History Introduced by Babson and Steingr´ ımsson (2000) in systematic search for Mahonian permutation statistics Selected other work:

Claesson – results for length 3 patterns with two adjacent letters (2001) Elizalde and Noy – consecutive patterns (2003) Elizalde – asymptotic enumeration (2006) Further enumerative results by Bernini, Ferrari, Kitaev, Mansour, Pergola, Pinzani

Survey article by Steingr´ ımsson in PP2007 proceedings

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Enumeration schemes for dashed permutation patterns Lara Pudwell Dashed Patterns

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Enumeration Scheme Definition/Notation Definition (informal) An enumeration scheme is an encoding for a family of recurrence relations enumerating members of a family of sets. Notation: Prefix Pattern Sn(Q)[p] = {π ∈ Sn(Q)|π1 · · · π|p| ∼ p} Note Sn(Q) = Sn(Q)[1] = Sn(Q)[12] ∪ Sn(Q)[21] = · · ·

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Enumeration schemes for dashed permutation patterns Lara Pudwell Dashed Patterns

Definition Brief History

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Enumeration Scheme Notation Notation: Children The children of prefix p ∈ Sn are the length n + 1 permutations whose first n letters are order-isomorphic to p. For example, the children of 213 are {2134, 2143, 3142, 3241}. Note Sn(Q)[p] =

  • c∈children(p)

Sn(Q)[c]

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Enumeration schemes for dashed permutation patterns Lara Pudwell Dashed Patterns

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Enumeration Scheme Notation Notation: Prefix Pattern with Specified Letters Sn(Q)[p; w] = {π ∈ Sn(Q)[p]|π1 · · · π|p| = w} Note

  • Sn(Q)[1]
  • =

n

  • a=1
  • Sn(Q)[1; a]
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Enumeration schemes for dashed permutation patterns Lara Pudwell Dashed Patterns

Definition Brief History

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Definition & Notation Example Algorithm Usefulness

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Enumeration Scheme Definitions Definition: Deletion Map dR For a set of indices R, dR(π) is the permutation obtained by deleting πr for all r ∈ R and reducing. For example, d{1,3}(53241) = red(341) = 231. Definition: Reversibly Deletable A set of indices R is reversibly deletable for prefix p with respect to Q if dR : Sn(Q)[p; w] → Sn−|R|(Q)[dR(p); dR(w)] is a bijection for all w such that Sn(Q)[p; w] = ∅.

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Enumeration schemes for dashed permutation patterns Lara Pudwell Dashed Patterns

Definition Brief History

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Enumeration Scheme Example Avoiding {23−1} Sn({23 − 1}) = Sn({23 − 1})[1] = Sn({23 − 1})[12]∪Sn({23 − 1})[21] Consider a member of Sn({23−1})[21]...

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Enumeration schemes for dashed permutation patterns Lara Pudwell Dashed Patterns

Definition Brief History

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Definition & Notation Example Algorithm Usefulness

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Enumeration Scheme Example Avoiding {23−1} Sn({23 − 1}) = Sn({23 − 1})[1] = Sn({23 − 1})[12]∪Sn({23 − 1})[21] Consider a member of Sn({23−1})[21]...

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Enumeration schemes for dashed permutation patterns Lara Pudwell Dashed Patterns

Definition Brief History

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Definition & Notation Example Algorithm Usefulness

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Enumeration Scheme Example Avoiding {23−1} Sn({23 − 1}) = Sn({23 − 1})[1] = Sn({23 − 1})[12]∪Sn({23 − 1})[21] Consider a member of Sn({23−1})[21]... So

  • Sn({23−1})[21; ab]
  • =
  • Sn−1({23−1})[1; b]
  • (i.e. {1} is reversibly deletable for p = 21, Q = {23−1}.)
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Enumeration schemes for dashed permutation patterns Lara Pudwell Dashed Patterns

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Two More Definitions... Definition: Spacing Vector Given a word w ∈ [n]k, let ci be the ith smallest letter of w, c0 = 0 and ck+1 = n + 1. The vector

  • g(n, w) = c1 − c0 − 1, c2 − c1 − 1, . . . , ck+1 − ck − 1

is the spacing vector of w. For example, w = 16372 ∈ [8]5 has c0 = 0, c1 = 1, c2 = 2, c3 = 3, c4 = 6, c5 = 7, c6 = 9. Thus, g(8, 16372) = 0, 0, 0, 2, 0, 1. Definition: Gap Vector

  • v ∈ Nk+1 is a gap vector for prefix p with respect to set of

patterns Q if for all n, Sn(Q)[p; w] = ∅ for all w such that

  • g(n, w) ≥

v componentwise.

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Enumeration schemes for dashed permutation patterns Lara Pudwell Dashed Patterns

Definition Brief History

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Definition & Notation Example Algorithm Usefulness

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Enumeration Scheme Example Avoiding {23−1} Sn({23 − 1}) = Sn({23 − 1})[1] = Sn({23 − 1})[12]∪Sn({23 − 1})[21] Consider a member of Sn({23−1})[12]...

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Enumeration schemes for dashed permutation patterns Lara Pudwell Dashed Patterns

Definition Brief History

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Definition & Notation Example Algorithm Usefulness

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Enumeration Scheme Example Avoiding {23−1} Sn({23 − 1}) = Sn({23 − 1})[1] = Sn({23 − 1})[12]∪Sn({23 − 1})[21] Consider a member of Sn({23−1})[12]... (1, 0, 0 is a gap vector for p = 12, Q = {23−1})

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Enumeration schemes for dashed permutation patterns Lara Pudwell Dashed Patterns

Definition Brief History

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Definition & Notation Example Algorithm Usefulness

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Enumeration Scheme Example Avoiding {23−1} Sn({23 − 1}) = Sn({23 − 1})[1] = Sn({23 − 1})[12]∪Sn({23 − 1})[21] Consider a member of Sn({23−1})[12]... (1, 0, 0 is a gap vector for p = 12, Q = {23−1}) So

  • Sn({23−1})[12; ab]
  • =
  • a > 1
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Enumeration schemes for dashed permutation patterns Lara Pudwell Dashed Patterns

Definition Brief History

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Enumeration Scheme Example Avoiding {23−1} Sn({23 − 1}) = Sn({23 − 1})[1] = Sn({23 − 1})[12]∪Sn({23 − 1})[21] Consider a member of Sn({23−1})[12]... (1, 0, 0 is a gap vector for p = 12, Q = {23−1}) So

  • Sn({23−1})[12; ab]
  • =
  • a > 1
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Enumeration schemes for dashed permutation patterns Lara Pudwell Dashed Patterns

Definition Brief History

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Definition & Notation Example Algorithm Usefulness

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Guaranteed Schemes Success Rates Sequences

Enumeration Scheme Example Avoiding {23−1} Sn({23 − 1}) = Sn({23 − 1})[1] = Sn({23 − 1})[12]∪Sn({23 − 1})[21] Consider a member of Sn({23−1})[12]... (1, 0, 0 is a gap vector for p = 12, Q = {23−1}) So

  • Sn({23−1})[12; ab]
  • =
  • a > 1
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Enumeration schemes for dashed permutation patterns Lara Pudwell Dashed Patterns

Definition Brief History

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Enumeration Scheme Example Avoiding {23−1} Sn({23 − 1}) = Sn({23 − 1})[1] = Sn({23 − 1})[12]∪Sn({23 − 1})[21] Consider a member of Sn({23−1})[12]... (1, 0, 0 is a gap vector for p = 12, Q = {23−1}) So

  • Sn({23−1})[12; ab]
  • =
  • a > 1
  • Sn−1({23−1})[1; b − 1]
  • a = 1
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Enumeration schemes for dashed permutation patterns Lara Pudwell Dashed Patterns

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Avoiding {23−1} Summary:

  • Sn({23−1})
  • =

n

  • a=1
  • Sn({23−1})[1; a]
  • Sn({23−1})[1; a]
  • =

i−1

  • b=1
  • Sn({23−1})[21; ab]
  • +

n

  • b=a+1
  • Sn({23−1})[12; ab]
  • Sn({23−1})[21; ab]
  • =
  • Sn−1({23−1})[1; b]
  • Sn({23−1})[12; ab]
  • =
  • a > 1
  • Sn−1({23−1})[1; b − 1]
  • a = 1
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Enumeration schemes for dashed permutation patterns Lara Pudwell Dashed Patterns

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Avoiding {23−1} Summary:

  • Sn({23−1})
  • =

n

  • a=1
  • Sn({23−1})[1; a]
  • Sn({23−1})[1; a]
  • =

i−1

  • b=1
  • Sn({23−1})[21; ab]
  • +

n

  • b=a+1
  • Sn({23−1})[12; ab]
  • Sn({23−1})[21; ab]
  • =
  • Sn−1({23−1})[1; b]
  • Sn({23−1})[12; ab]
  • =
  • a > 1
  • Sn−1({23−1})[1; b − 1]
  • a = 1

|Sn({23−1})| = Bn and |Sn({23−1})[1; a]| is the number of set partitions where a is the largest element in the block containing 1.

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Enumeration Scheme “Algorithm”

1

Let P = {1}

2

Let P∗ be the set of all children of p ∈ P

3

For p∗ ∈ P∗:

Compute the set of minimal gap vectors for p∗. Compute the set of reversibly deletable indices R for p∗.

4

Let P be the set of prefixes in P∗ for which R = ∅ in step

  • 3. If P = ∅ then return to step 2; otherwise end.
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Enumeration schemes for dashed permutation patterns Lara Pudwell Dashed Patterns

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Usefulness of Enumeration Schemes Discovery can be completely automated by computer. Can sometimes be transformed into a functional equation satisfied by generating function. (Baxter 2011) Can be used to q-count according to inversion number, and according to the number of occurrences of any consecutive pattern(s). (Baxter 2011) Extend to many other types of pattern avoidance

  • problems. (classical avoidance, pattern-avoiding words,

barred patterns)

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Guaranteed Schemes Theorem 1 – Consecutive Patterns If σ is a dashless pattern of length t, then {σ} has a finite enumeration scheme of depth t. Complements work of Aldred et al, Ehrenborg et al, Nakamura, Remmel et al, and others

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Enumeration schemes for dashed permutation patterns Lara Pudwell Dashed Patterns

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Guaranteed Schemes Theorem 1 – Consecutive Patterns If σ is a dashless pattern of length t, then {σ} has a finite enumeration scheme of depth t. Theorem 2 – “Nearly Consecutive” Patterns If σ is a pattern of length t where only the last two numbers have a dash between them, then {σ} has a finite enumeration scheme of depth t − 1. Complements work of Elizalde, Kitaev, and others.

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Enumeration schemes for dashed permutation patterns Lara Pudwell Dashed Patterns

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Guaranteed Schemes Theorem 1 – Consecutive Patterns If σ is a dashless pattern of length t, then {σ} has a finite enumeration scheme of depth t. Theorem 2 – “Nearly Consecutive” Patterns If σ is a pattern of length t where only the last two numbers have a dash between them, then {σ} has a finite enumeration scheme of depth t − 1. Theorem 3 If the finite set Q contains only consecutive and “nearly consecutive” patterns, then Q has a finite enumeration scheme.

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Enumeration schemes for dashed permutation patterns Lara Pudwell Dashed Patterns

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Dashes vs. Depth Question Is number of dashes related to scheme depth? σ Depth 1234 4 123 − 4 3 12 − 34 4 1 − 234 4 12 − 3 − 4 4 1 − 23 − 4 3 1 − 2 − 34 5 1 − 2 − 3 − 4 4

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Enumeration schemes for dashed permutation patterns Lara Pudwell Dashed Patterns

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Dashes vs. Depth Question Is number of dashes related to scheme depth? σ Depth 1234 4 123 − 4 3 12 − 34 4 1 − 234 4 12 − 3 − 4 4 1 − 23 − 4 3 1 − 2 − 34 5 1 − 2 − 3 − 4 4 Answer: Maybe, but not monotonically.

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Terminology Trivial Symmetries Note: |Sn(Q)| = |Sn(Qc)| = |Sn(Qr)| where Qc is the set of complements of patterns in Q and Qr is the set of reversals of patterns in Q. (d,w)-Scheme Countable A pattern set Q is (d,w)-Scheme Countable if it has an enumeration scheme of depth ≤ d where all gap vectors have weight | v| = v1 + · · · v|

v| ≤ w.

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Enumeration schemes for dashed permutation patterns Lara Pudwell Dashed Patterns

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Success Rate by Block Type

Block Type The block type of a dashed pattern is a vector describing the number

  • f letters between each dash.

Block type # Trivial # (5,2)-Scheme % symmetry classes Countable classes (3) 2 2 100% (2,1) 3 3 100% (1,1,1) 2 2 100% (4) 8 8 100% (3,1) 12 12 100% (2,2) 8 3 37.5% (2,1,1) 12 4 25% (1,2,1) 8 6 75% (1,1,1,1) 7 2 28.6% (5) 32 32 100% (4,1) 32 32 100%

  • ther length 5

415 80 19.3%

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Success Rate by Pattern Set

Set Type The set type of a set of patterns Q is the multiset {|σ| : σ ∈ Q}. Set type # Trivial # (5,2)-Scheme % symmetry classes Countable classes {2} 2 2 100% {2,2} 3 3 100% {2,3} 11 11 100% {3} 7 7 100% {3,3} 70 68 97.1% {3,3,3} 358 354 98.9% {4} 55 35 63.6% {4,4} 4624 1600 34.6% {5} 479 144 30.1% {3,4} 914 639 69.9% {3,5} 7411 2465 33.3%

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Sequences for Length 4 Patterns

σ {|Sn({σ})|}n OEIS No. 1 − 24 − 3, 1 − 42 − 3 1, 2, 6, 23, 104, 532, 3004,. . . A137538 1 − 23 − 4, 1 − 32 − 4 1, 2, 6, 23, 105, 549, 3207,. . . A113227 1 − 34 − 2, 1 − 43 − 2 12 − 3 − 4, 12 − 4 − 3 1, 2, 6, 23, 105, 550, 3228,. . . New 21 − 3 − 4, 21 − 4 − 3 143 − 2 1, 2, 6, 23, 107, 582, 3622,. . . New 214 − 3 1, 2, 6, 23, 107, 583, 3637,. . . New 124 − 3, 421 − 3 1, 2, 6, 23, 107, 584, 3660,. . . New 12 − 34, 12 − 43, 21 − 43 1, 2, 6, 23, 107, 585, 3669,. . . A113226 132 − 4, 231 − 4, 312 − 4, 1, 2, 6, 23, 107, 585, 3671,. . . A071075 213 − 4, 142 − 3, 241 − 3 123 − 4, 321 − 4 1, 2, 6, 23, 108, 598, 3815,. . . A071076

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Interesting? Sequences

Q OEIS No. Description {13−2, 213} A105633 Number of Dyck paths

  • f semilength n + 1

avoiding UUDU {1−2−3, 231} A135307 Number of Dyck paths

  • f semilength n

avoiding UDDU {12−3, 1−3−2, 312} A005314 Number of compositions

  • f n into parts

congruent to {1, 2} mod 4 {1−2−3, 231, 3−1−2} A089071 Number of liberties a big eye of size n gives in the game of Go

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Recap Many results for dashed patterns over past decade. Now have algorithm to compute enumeration scheme recurrences for sets of dashed patterns. Guaranteed success for consecutive or “nearly consecutive” patterns. Confirmed many known results, generated many new results and conjectures.

Wilf equivalences, enumeration results, connections to

  • ther combinatorial sequences in OEIS
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Thank You!