Heaps, periodic parallelogram polyominoes, and 321-avoiding affine - - PowerPoint PPT Presentation

FC elements 321-avoiding elements Parellolograms polyominoes Generating functions

Heaps, periodic parallelogram polyominoes, and 321-avoiding affine permutations

Riccardo Biagioli (Universit´ e Lyon 1) joint work with

• M. Bousquet-M´

elou, F. Jouhet, P. Nadeau GASCOM 2016 03 juin 2016

Riccardo Biagioli (Universit´ e Lyon 1) joint work with M. Bousquet-M´ elou, F. Jouhet, P. Nadeau Heaps, periodic parallelogram polyominoes, and 321-avoiding affine permutations

FC elements 321-avoiding elements Parellolograms polyominoes Generating functions

Outline of the talk

1 Motivations 2 321-avoiding permutations and alternating diagrams 3 Periodic parallelogram polyominoes 4 Marked heaps of segments 5 Computations of the corresponding generating functions Riccardo Biagioli (Universit´ e Lyon 1) joint work with M. Bousquet-M´ elou, F. Jouhet, P. Nadeau Heaps, periodic parallelogram polyominoes, and 321-avoiding affine permutations

FC elements 321-avoiding elements Parellolograms polyominoes Generating functions

Reduced decompositions

The symmetric group Sn is generated as a Coxeter group by the set S of simple transpositions si = (i, i + 1) with Relations:      s2 = 1 sisi+1si = si+1sisi+1 (braid relations) sisj = sjsi if j = i ± 1 (commutation relations) Definition (Length) ℓ(w) = minimal l such that w = s1s2 · · · sl with si ∈ S Such a minimal word is a reduced decomposition of w. Proposition (Matsumoto-Tits property) Given two reduced decompositions of w, there is a sequence of braid or commutation relations which can be applied to transform

• ne into the other.

Riccardo Biagioli (Universit´ e Lyon 1) joint work with M. Bousquet-M´ elou, F. Jouhet, P. Nadeau Heaps, periodic parallelogram polyominoes, and 321-avoiding affine permutations

FC elements 321-avoiding elements Parellolograms polyominoes Generating functions

Reduced decompositions

The symmetric group Sn is generated as a Coxeter group by the set S of simple transpositions si = (i, i + 1) with Relations:      s2 = 1 sisi+1si = si+1sisi+1 (braid relations) sisj = sjsi if j = i ± 1 (commutation relations) Definition (Length) ℓ(w) = minimal l such that w = s1s2 · · · sl with si ∈ S Such a minimal word is a reduced decomposition of w. Proposition (Matsumoto-Tits property) Given two reduced decompositions of w, there is a sequence of braid or commutation relations which can be applied to transform

• ne into the other.

Riccardo Biagioli (Universit´ e Lyon 1) joint work with M. Bousquet-M´ elou, F. Jouhet, P. Nadeau Heaps, periodic parallelogram polyominoes, and 321-avoiding affine permutations

FC elements 321-avoiding elements Parellolograms polyominoes Generating functions

Reduced decompositions

The symmetric group Sn is generated as a Coxeter group by the set S of simple transpositions si = (i, i + 1) with Relations:      s2 = 1 sisi+1si = si+1sisi+1 (braid relations) sisj = sjsi if j = i ± 1 (commutation relations) Definition (Length) ℓ(w) = minimal l such that w = s1s2 · · · sl with si ∈ S Such a minimal word is a reduced decomposition of w. Proposition (Matsumoto-Tits property) Given two reduced decompositions of w, there is a sequence of braid or commutation relations which can be applied to transform

• ne into the other.

Riccardo Biagioli (Universit´ e Lyon 1) joint work with M. Bousquet-M´ elou, F. Jouhet, P. Nadeau Heaps, periodic parallelogram polyominoes, and 321-avoiding affine permutations

FC elements 321-avoiding elements Parellolograms polyominoes Generating functions

Fully commutative elements

Definition An element w is fully commutative if given two reduced decompositions of w, there is a sequence of commutation relations which can be applied to transform one into the other. In general, the set Red(w) of reduced decompositions of w splits into several commutation classes: w is fully commutative if there is only one such class. Red(w) =

C1 C2 C4 C3 Riccardo Biagioli (Universit´ e Lyon 1) joint work with M. Bousquet-M´ elou, F. Jouhet, P. Nadeau Heaps, periodic parallelogram polyominoes, and 321-avoiding affine permutations

FC elements 321-avoiding elements Parellolograms polyominoes Generating functions

Fully commutative elements

Definition An element w is fully commutative if given two reduced decompositions of w, there is a sequence of commutation relations which can be applied to transform one into the other. In general, the set Red(w) of reduced decompositions of w splits into several commutation classes: w is fully commutative if there is only one such class. Red(w) =

C1 C2 C4 C3 Riccardo Biagioli (Universit´ e Lyon 1) joint work with M. Bousquet-M´ elou, F. Jouhet, P. Nadeau Heaps, periodic parallelogram polyominoes, and 321-avoiding affine permutations

FC elements 321-avoiding elements Parellolograms polyominoes Generating functions

Fully commutative elements

Definition An element w is fully commutative if given two reduced decompositions of w, there is a sequence of commutation relations which can be applied to transform one into the other. In general, the set Red(w) of reduced decompositions of w splits into several commutation classes: w is fully commutative if there is only one such class. Red(w) =

C1

then w is FC

Riccardo Biagioli (Universit´ e Lyon 1) joint work with M. Bousquet-M´ elou, F. Jouhet, P. Nadeau Heaps, periodic parallelogram polyominoes, and 321-avoiding affine permutations

FC elements 321-avoiding elements Parellolograms polyominoes Generating functions

Previous works

[Billey–Jockush–Stanley (1993)] 321-avoiding permutations in Sn correspond to fully commutative elements of type An−1 [Barcucci–Del Lungo–Pergola–Pinzani (2001)] enumeration of Sn(321) with respect to the inversions: nice expression. [Green (2001)] 321-avoiding affine permutations in Sn correspond to fully commutative elements of type An−1. [Hanusa–Jones (2010)] enumeration of Sn(321) with respect to the inversions (or Coxeter length): nice periodicity properties but very complicated expression for the GF. [Biagioli–Jouhet–Nadeau (2014)] enumeration of Sn(321) and

• Sn(321) with respect to the inversions using alternating

diagrams and Motzkin-type lattice walks: recursive GF.

Riccardo Biagioli (Universit´ e Lyon 1) joint work with M. Bousquet-M´ elou, F. Jouhet, P. Nadeau Heaps, periodic parallelogram polyominoes, and 321-avoiding affine permutations

FC elements 321-avoiding elements Parellolograms polyominoes Generating functions

Enumeration of FC elements by length

Question Enumerate FC elements by length for any finite or affine Coxeter group W ? Compute the generating series W FC(q) =

• w∈FC

qℓ(w) and W FC(q, x) =

• n≥0

W FC(q)xn Theorem (B., Jouhet, Nadeau, 2014) We computed W FC(q) for any finite and affine W . If W is affine, the coefficients of W FC(q) form an ultimately periodic sequence. The main tool is to encode FC elements by certain lattice paths. BUT this does not give rise to nice expressions for W FC(q, x).

Riccardo Biagioli (Universit´ e Lyon 1) joint work with M. Bousquet-M´ elou, F. Jouhet, P. Nadeau Heaps, periodic parallelogram polyominoes, and 321-avoiding affine permutations

FC elements 321-avoiding elements Parellolograms polyominoes Generating functions

Enumeration of FC elements by length

Question Enumerate FC elements by length for any finite or affine Coxeter group W ? Compute the generating series W FC(q) =

• w∈FC

qℓ(w) and W FC(q, x) =

• n≥0

W FC(q)xn Theorem (B., Jouhet, Nadeau, 2014) We computed W FC(q) for any finite and affine W . If W is affine, the coefficients of W FC(q) form an ultimately periodic sequence. The main tool is to encode FC elements by certain lattice paths. BUT this does not give rise to nice expressions for W FC(q, x).

Riccardo Biagioli (Universit´ e Lyon 1) joint work with M. Bousquet-M´ elou, F. Jouhet, P. Nadeau Heaps, periodic parallelogram polyominoes, and 321-avoiding affine permutations

FC elements 321-avoiding elements Parellolograms polyominoes Generating functions

An example

Generating functions AFC

n−1(q): the first ones are

• AFC

2 (q) = 1 + 3q + 6q2 + 6q3 + 6q4 + · · ·

• AFC

3 (q) = 1 + 4q + 10q2 + 16q3 + 18q4 + 16q5 + 18q6 + · · ·

• AFC

4 (q) = 1 + 5q + 15q2 + 30q3 + 45q4

+50q5 + 50q6 + 50q7 + 50q8 + 50q9 + · · ·

• AFC

5 (q) = 1 + 6q + 21q2 + 50q3 + 90q4 + 126q5 + 146q6

+150q7 + 156q8 + 152q9 + 156q10 + 150q11 + 158q12 +150q13 + 156q14 + 152q15 + 156q16 + 150q17 + 158q18 + · · · The coefficients of AFC

n−1(q) are ultimately periodic of period dividing n,

and the periodicity starts from degree 1 + ⌈(n − 1)/2⌉⌊(n + 1)/2⌋.

Riccardo Biagioli (Universit´ e Lyon 1) joint work with M. Bousquet-M´ elou, F. Jouhet, P. Nadeau Heaps, periodic parallelogram polyominoes, and 321-avoiding affine permutations

FC elements 321-avoiding elements Parellolograms polyominoes Generating functions

Generating function for permutations in Sn(321)

An−1(q) :=

• σ∈Sn(321)

qinv(σ) and A(x) =

• n≥0

AFC

n (q)xn.

Theorem (Barcucci, Del Lungo, Pergola, Pinzani (2001)) We have A(x) = 1 1 − xq × J(xq) J(x) , where J(x) :=

• n≥0

(−x)nq(n

2)

(q)n(xq)n Here (a)0 := 1 and (a)n := (1 − a)(1 − aq) · · · (1 − aqn−1), n ≥ 1

Riccardo Biagioli (Universit´ e Lyon 1) joint work with M. Bousquet-M´ elou, F. Jouhet, P. Nadeau Heaps, periodic parallelogram polyominoes, and 321-avoiding affine permutations

FC elements 321-avoiding elements Parellolograms polyominoes Generating functions

Generating function for affine permutations in ˜ Sn(321)

• An−1(q) :=
• σ∈

Sn(321)

qinv(σ) and A(x) :=

• n≥1
• An−1(q)xn

Theorem (B., Bousquet-M´ elou, Jouhet, Nadeau (2015)) We have

• A(x) = −x J′(x)

J(x) −

• n≥1

xnqn 1 − qn Our strategy: encode 321-avoiding (affine) permutations by alternating (affine) diagrams, then by (periodic) parallelogram polyominoes, and finally by heaps of segments.

Riccardo Biagioli (Universit´ e Lyon 1) joint work with M. Bousquet-M´ elou, F. Jouhet, P. Nadeau Heaps, periodic parallelogram polyominoes, and 321-avoiding affine permutations

FC elements 321-avoiding elements Parellolograms polyominoes Generating functions

Generating function for affine permutations in ˜ Sn(321)

• An−1(q) :=
• σ∈

Sn(321)

qinv(σ) and A(x) :=

• n≥1
• An−1(q)xn

Theorem (B., Bousquet-M´ elou, Jouhet, Nadeau (2015)) We have

• A(x) = −x J′(x)

J(x) −

• n≥1

xnqn 1 − qn Our strategy: encode 321-avoiding (affine) permutations by alternating (affine) diagrams, then by (periodic) parallelogram polyominoes, and finally by heaps of segments.

Riccardo Biagioli (Universit´ e Lyon 1) joint work with M. Bousquet-M´ elou, F. Jouhet, P. Nadeau Heaps, periodic parallelogram polyominoes, and 321-avoiding affine permutations

FC elements 321-avoiding elements Parellolograms polyominoes Generating functions

321-avoiding permutations

A permutation σ ∈ Sn is 321-avoiding if no integers i < j < k are such that σ(i) > σ(j) > σ(k) In S6, σ = 513624 is not 321-avoiding while σ = 231564 is. They are counted by Catalan numbers 1 n + 1 2n n

• .

The inversion number inv(σ) is the number of inversions of the permutation σ i.e. inv(σ) = {(i, j) ∈ [n]2 | i < j and σ(i) > σ(j)}. It is well known that ℓ(σ) = inv(σ) for any σ ∈ Sn.

Riccardo Biagioli (Universit´ e Lyon 1) joint work with M. Bousquet-M´ elou, F. Jouhet, P. Nadeau Heaps, periodic parallelogram polyominoes, and 321-avoiding affine permutations

FC elements 321-avoiding elements Parellolograms polyominoes Generating functions

From line diagrams to alternating diagrams

Take the 321-avoiding permutation σ = 461279358 ∈ S9.

1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9

Riccardo Biagioli (Universit´ e Lyon 1) joint work with M. Bousquet-M´ elou, F. Jouhet, P. Nadeau Heaps, periodic parallelogram polyominoes, and 321-avoiding affine permutations

FC elements 321-avoiding elements Parellolograms polyominoes Generating functions

From line diagrams to alternating diagrams

Take the 321-avoiding permutation σ = 461279358 ∈ S9.

1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9

Riccardo Biagioli (Universit´ e Lyon 1) joint work with M. Bousquet-M´ elou, F. Jouhet, P. Nadeau Heaps, periodic parallelogram polyominoes, and 321-avoiding affine permutations

FC elements 321-avoiding elements Parellolograms polyominoes Generating functions

From line diagrams to alternating diagrams

Take the 321-avoiding permutation σ = 461279358 ∈ S9.

1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9

Riccardo Biagioli (Universit´ e Lyon 1) joint work with M. Bousquet-M´ elou, F. Jouhet, P. Nadeau Heaps, periodic parallelogram polyominoes, and 321-avoiding affine permutations

FC elements 321-avoiding elements Parellolograms polyominoes Generating functions

Alternating diagrams

Take the 321-avoiding permutation σ = 461279358 ∈ S9.

1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9

Riccardo Biagioli (Universit´ e Lyon 1) joint work with M. Bousquet-M´ elou, F. Jouhet, P. Nadeau Heaps, periodic parallelogram polyominoes, and 321-avoiding affine permutations

FC elements 321-avoiding elements Parellolograms polyominoes Generating functions

From line diagrams to alternating diagrams

Take the 321-avoiding permutation σ = 461279358 ∈ S9.

1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9

Riccardo Biagioli (Universit´ e Lyon 1) joint work with M. Bousquet-M´ elou, F. Jouhet, P. Nadeau Heaps, periodic parallelogram polyominoes, and 321-avoiding affine permutations

FC elements 321-avoiding elements Parellolograms polyominoes Generating functions

From line diagrams to alternating diagrams

Take the 321-avoiding permutation σ = 461279358 ∈ S9.

1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 s1 s2 s3 s4 s5 s6 s7 s8

The number of intersections is the inversion number of σ. Each intersection point is associated with a generator of Sn. By reading the generators from left to right and bottom to top we obtain a reduced expression for σ. Infact we can read all of them from this diagram !

Riccardo Biagioli (Universit´ e Lyon 1) joint work with M. Bousquet-M´ elou, F. Jouhet, P. Nadeau Heaps, periodic parallelogram polyominoes, and 321-avoiding affine permutations

FC elements 321-avoiding elements Parellolograms polyominoes Generating functions

From line diagrams to alternating diagrams

Take the 321-avoiding permutation σ = 461279358 ∈ S9.

1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 s1 s2 s3 s4 s5 s6 s7 s8

The number of intersections is the inversion number of σ. Each intersection point is associated with a generator of Sn. By reading the generators from left to right and bottom to top we obtain a reduced expression for σ. Infact we can read all of them from this diagram !

Riccardo Biagioli (Universit´ e Lyon 1) joint work with M. Bousquet-M´ elou, F. Jouhet, P. Nadeau Heaps, periodic parallelogram polyominoes, and 321-avoiding affine permutations

FC elements 321-avoiding elements Parellolograms polyominoes Generating functions

From line diagrams to alternating diagrams

Take the 321-avoiding permutation σ = 461279358 ∈ S9.

1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 s1 s2 s3 s4 s5 s6 s7 s8

The number of intersections is the inversion number of σ. Each intersection point is associated with a generator of Sn. By reading the generators from left to right and bottom to top we obtain a reduced expression for σ. Infact we can read all of them from this diagram !

Riccardo Biagioli (Universit´ e Lyon 1) joint work with M. Bousquet-M´ elou, F. Jouhet, P. Nadeau Heaps, periodic parallelogram polyominoes, and 321-avoiding affine permutations

FC elements 321-avoiding elements Parellolograms polyominoes Generating functions

From line diagrams to alternating diagrams

Take the 321-avoiding permutation σ = 461279358 ∈ S9.

1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 s1 s2 s3 s4 s5 s6 s7 s8

The number of intersections is the inversion number of σ. Each intersection point is associated with a generator of Sn. By reading the generators from left to right and bottom to top we obtain a reduced expression for σ. Infact we can read all of them from this diagram !

Riccardo Biagioli (Universit´ e Lyon 1) joint work with M. Bousquet-M´ elou, F. Jouhet, P. Nadeau Heaps, periodic parallelogram polyominoes, and 321-avoiding affine permutations

FC elements 321-avoiding elements Parellolograms polyominoes Generating functions

Characterization of alternating diagrams

The alternating diagrams avoids precisely such configutations:

si si+1 si+2 ∅ si si+1 si+2 ∅ si

Proposition Alternating diagrams of Sn are characterized by: (a) At most one occurrence of s1 (resp. sn−1) (b) ∀i, elements with labels si, si+1 form an alternating chain

Riccardo Biagioli (Universit´ e Lyon 1) joint work with M. Bousquet-M´ elou, F. Jouhet, P. Nadeau Heaps, periodic parallelogram polyominoes, and 321-avoiding affine permutations

FC elements 321-avoiding elements Parellolograms polyominoes Generating functions

321-avoiding affine permutations

Definition (Affine permutations) The group Sn is the set of permutations σ of Z satisfying σ(i + n) = σ(i) + n, and n

i=1 σ(i) = n i=1 i.

. . . , | −2, −11, −9, 0 |, 2, −7, −5, 4, | 6, −3, −1, 8, | 10, 1, 3, 12, | 14, 5, 7, 16, . . . Here σ ∈ S4(321) and inv(σ) = 5 + 4 = 9

1 2 3 4 s0 s1 s2 s3 s0 1 2 3 4 −3 −1 6 8 −3 6 7

The element σ = [6, −3, −1, 8] = s1s3s0s2s1s3s0s2s1.

Riccardo Biagioli (Universit´ e Lyon 1) joint work with M. Bousquet-M´ elou, F. Jouhet, P. Nadeau Heaps, periodic parallelogram polyominoes, and 321-avoiding affine permutations

FC elements 321-avoiding elements Parellolograms polyominoes Generating functions

321-avoiding affine permutations

Definition (Affine permutations) The group Sn is the set of permutations σ of Z satisfying σ(i + n) = σ(i) + n, and n

i=1 σ(i) = n i=1 i.

. . . , | −2, −11, −9, 0 |, 2, −7, −5, 4, | 6, −3, −1, 8, | 10, 1, 3, 12, | 14, 5, 7, 16, . . . Here σ ∈ S4(321) and inv(σ) = 5 + 4 = 9

1 2 3 4 s0 s1 s2 s3 s0 1 2 3 4 −3 −1 6 8 −3 6 7

The element σ = [6, −3, −1, 8] = s1s3s0s2s1s3s0s2s1.

Riccardo Biagioli (Universit´ e Lyon 1) joint work with M. Bousquet-M´ elou, F. Jouhet, P. Nadeau Heaps, periodic parallelogram polyominoes, and 321-avoiding affine permutations

FC elements 321-avoiding elements Parellolograms polyominoes Generating functions

Affine alternating diagrams (on a cylinder)

Same local conditions as for Sn: must avoid the shapes

si si+1 si+2 ∅ si si+1 si+2 ∅ si

Difference: the labels above must be taken with index modulo n; the posets must be thought of as “drawn on a cylinder”

the same

s0 s1 s2 s3 s4 s5 s0

Riccardo Biagioli (Universit´ e Lyon 1) joint work with M. Bousquet-M´ elou, F. Jouhet, P. Nadeau Heaps, periodic parallelogram polyominoes, and 321-avoiding affine permutations

FC elements 321-avoiding elements Parellolograms polyominoes Generating functions

From alternating diagrams to parellelogram polyominoes

Classical case (Viennot)

(1, 5), (5, 5), (4, 4), (1, 1), (1, 3), (2, 3) Convention: a1 = 1

Riccardo Biagioli (Universit´ e Lyon 1) joint work with M. Bousquet-M´ elou, F. Jouhet, P. Nadeau Heaps, periodic parallelogram polyominoes, and 321-avoiding affine permutations

FC elements 321-avoiding elements Parellolograms polyominoes Generating functions

From alternating diagrams to parellelogram polyominoes

Classical case (Viennot)

(1, 5), (5, 5), (4, 4), (1, 1), (1, 3), (2, 3) Convention: a1 = 1

Riccardo Biagioli (Universit´ e Lyon 1) joint work with M. Bousquet-M´ elou, F. Jouhet, P. Nadeau Heaps, periodic parallelogram polyominoes, and 321-avoiding affine permutations

FC elements 321-avoiding elements Parellolograms polyominoes Generating functions

From alternating diagrams to parellelogram polyominoes

Classical case (Viennot)

(1, 5), (5, 5), (4, 4), (1, 1), (1, 3), (2, 3) Convention: a1 = 1

Riccardo Biagioli (Universit´ e Lyon 1) joint work with M. Bousquet-M´ elou, F. Jouhet, P. Nadeau Heaps, periodic parallelogram polyominoes, and 321-avoiding affine permutations

FC elements 321-avoiding elements Parellolograms polyominoes Generating functions

From parallelogram polyominoes to heaps of segments

Definition (The set S) Let S be the set of finite sequences (ai, bi) satisfying a1 ≤ b1 ≥ a2 ≤ b2 ≥ . . . ≤ bn−1 ≥ an ≤ bn.

(1, 5), (5, 5), (4, 4), (1, 1), (1, 3), (2, 3) Convention: a1 = 1

Parallelogram polyominoes are coded by sequences in S with a1 = 1, where: bi is the height of the i-column Ci. ai is the number of cells of Ci in contact with column Ci−1.

Riccardo Biagioli (Universit´ e Lyon 1) joint work with M. Bousquet-M´ elou, F. Jouhet, P. Nadeau Heaps, periodic parallelogram polyominoes, and 321-avoiding affine permutations

FC elements 321-avoiding elements Parellolograms polyominoes Generating functions

From parallelogram polyominoes to heaps of segments

Definition (The set S) Let S be the set of finite sequences (ai, bi) satisfying a1 ≤ b1 ≥ a2 ≤ b2 ≥ . . . ≤ bn−1 ≥ an ≤ bn.

(1, 5), (5, 5), (4, 4), (1, 1), (1, 3), (2, 3) Convention: a1 = 1

Parallelogram polyominoes are coded by sequences in S with a1 = 1, where: bi is the height of the i-column Ci. ai is the number of cells of Ci in contact with column Ci−1.

Riccardo Biagioli (Universit´ e Lyon 1) joint work with M. Bousquet-M´ elou, F. Jouhet, P. Nadeau Heaps, periodic parallelogram polyominoes, and 321-avoiding affine permutations

FC elements 321-avoiding elements Parellolograms polyominoes Generating functions

From parallelogram polyominoes to heaps of segments

Classical case (Bousquet-M´ elou–Viennot)

(1, 5), (5, 5), (4, 4), (1, 1), (1, 3), (2, 3) Convention: a1 = 1 1 2 3 4 5 Semi pyramid: unique max [1, 5] f

Riccardo Biagioli (Universit´ e Lyon 1) joint work with M. Bousquet-M´ elou, F. Jouhet, P. Nadeau Heaps, periodic parallelogram polyominoes, and 321-avoiding affine permutations

FC elements 321-avoiding elements Parellolograms polyominoes Generating functions

From parallelogram polyominoes to heaps of segments

Classical case (Bousquet-M´ elou–Viennot)

(1, 5), (5, 5), (4, 4), (1, 1), (1, 3), (2, 3) Convention: a1 = 1 1 2 3 4 5 Semi pyramid: unique max [1, 5] f

Riccardo Biagioli (Universit´ e Lyon 1) joint work with M. Bousquet-M´ elou, F. Jouhet, P. Nadeau Heaps, periodic parallelogram polyominoes, and 321-avoiding affine permutations

FC elements 321-avoiding elements Parellolograms polyominoes Generating functions

From parallelogram polyominoes to heaps of segments

Classical case (Bousquet-M´ elou–Viennot)

(1, 5), (5, 5), (4, 4), (1, 1), (1, 3), (2, 3) Convention: a1 = 1 1 2 3 4 5 Semi pyramid: unique max [1, 5] f

Riccardo Biagioli (Universit´ e Lyon 1) joint work with M. Bousquet-M´ elou, F. Jouhet, P. Nadeau Heaps, periodic parallelogram polyominoes, and 321-avoiding affine permutations

FC elements 321-avoiding elements Parellolograms polyominoes Generating functions

From parallelogram polyominoes to heaps of segments

Classical case (Bousquet-M´ elou–Viennot)

(1, 5), (5, 5), (4, 4), (1, 1), (1, 3), (2, 3) Convention: a1 = 1 1 2 3 4 5 Semi pyramid: unique max [1, 5] f

Riccardo Biagioli (Universit´ e Lyon 1) joint work with M. Bousquet-M´ elou, F. Jouhet, P. Nadeau Heaps, periodic parallelogram polyominoes, and 321-avoiding affine permutations

FC elements 321-avoiding elements Parellolograms polyominoes Generating functions

From parallelogram polyominoes to heaps of segments

Classical case (Bousquet-M´ elou–Viennot)

(1, 5), (5, 5), (4, 4), (1, 1), (1, 3), (2, 3) Convention: a1 = 1 1 2 3 4 5 Semi pyramid: unique max [1, 5] f

Riccardo Biagioli (Universit´ e Lyon 1) joint work with M. Bousquet-M´ elou, F. Jouhet, P. Nadeau Heaps, periodic parallelogram polyominoes, and 321-avoiding affine permutations

FC elements 321-avoiding elements Parellolograms polyominoes Generating functions

From parallelogram polyominoes to heaps of segments

Classical case (Bousquet-M´ elou–Viennot)

(1, 5), (5, 5), (4, 4), (1, 1), (1, 3), (2, 3) Convention: a1 = 1 1 2 3 4 5 Semi pyramid: unique max [1, 5] f

Theorem More generally, the map f is a bijection between the set of sequences S and the set H of heaps of segments.

Riccardo Biagioli (Universit´ e Lyon 1) joint work with M. Bousquet-M´ elou, F. Jouhet, P. Nadeau Heaps, periodic parallelogram polyominoes, and 321-avoiding affine permutations

FC elements 321-avoiding elements Parellolograms polyominoes Generating functions

From affine alt diag to periodic parellelogram polyominoes

Affine case

(2, 4), (3, 5), (5, 5), (4, 4), (1, 1), (1, 3) A mark nedeed a mark 1 ≤ a1 ≤ bn to recover s0 The periodicity identifies a1

Riccardo Biagioli (Universit´ e Lyon 1) joint work with M. Bousquet-M´ elou, F. Jouhet, P. Nadeau Heaps, periodic parallelogram polyominoes, and 321-avoiding affine permutations

FC elements 321-avoiding elements Parellolograms polyominoes Generating functions

From affine alt diag to periodic parellelogram polyominoes

Affine case

(2, 4), (3, 5), (5, 5), (4, 4), (1, 1), (1, 3) A mark nedeed a mark 1 ≤ a1 ≤ bn to recover s0 The periodicity identifies a1

Riccardo Biagioli (Universit´ e Lyon 1) joint work with M. Bousquet-M´ elou, F. Jouhet, P. Nadeau Heaps, periodic parallelogram polyominoes, and 321-avoiding affine permutations

FC elements 321-avoiding elements Parellolograms polyominoes Generating functions

From affine alt diag to periodic parellelogram polyominoes

Affine case

(2, 4), (3, 5), (5, 5), (4, 4), (1, 1), (1, 3) A mark nedeed a mark 1 ≤ a1 ≤ bn to recover s0 The periodicity identifies a1

Riccardo Biagioli (Universit´ e Lyon 1) joint work with M. Bousquet-M´ elou, F. Jouhet, P. Nadeau Heaps, periodic parallelogram polyominoes, and 321-avoiding affine permutations

FC elements 321-avoiding elements Parellolograms polyominoes Generating functions

Periodic parallelogram polyominoes

Definition (PPP) A periodic parallelogram polyomino is a parallelogram polyomino marked in its first column in a, where a is an integer between 1 and the height of the last column. Periodic parellelogram polyominoes as sequences These naturally correspond to sequences (ai, bi)1≤i≤n ∈ S such that a1 ≤ bn, i.e. bn ≥ a1 ≤ b1 ≥ a2 ≤ b2 ≥ . . . ≤ bn−1 ≥ an ≤ bn.

Riccardo Biagioli (Universit´ e Lyon 1) joint work with M. Bousquet-M´ elou, F. Jouhet, P. Nadeau Heaps, periodic parallelogram polyominoes, and 321-avoiding affine permutations

FC elements 321-avoiding elements Parellolograms polyominoes Generating functions

From PPP to heaps of segments

Affine case

(2, 4), (3, 5), (5, 5), (4, 4), (1, 1), (1, 3) Periodicity 1 ≤ a1 ≤ bn 1 2 3 4 5 a1 b1 an bn Condition (I) on the heap

• n left min et right max elements

a1 bn f

Riccardo Biagioli (Universit´ e Lyon 1) joint work with M. Bousquet-M´ elou, F. Jouhet, P. Nadeau Heaps, periodic parallelogram polyominoes, and 321-avoiding affine permutations

FC elements 321-avoiding elements Parellolograms polyominoes Generating functions

A new family of heaps of segments

Definition (Condition (I)) Let H be a heap of segments. If [a, b] is the rightmost maximal segment of H and [a′, b′] is its leftmost minimal segment, then a ≤ b′. Let HI be the set of heaps satisfying (I). Proposition The map f induces a bijection between the set PPP and HI.

Riccardo Biagioli (Universit´ e Lyon 1) joint work with M. Bousquet-M´ elou, F. Jouhet, P. Nadeau Heaps, periodic parallelogram polyominoes, and 321-avoiding affine permutations

FC elements 321-avoiding elements Parellolograms polyominoes Generating functions

Marked PPP to marked heaps of segments in H∗

I

(2, 4), (3, 5), (5, 5), (4, 4), (1, 1), (1, 3) Mark in [a1, b1] 1 2 3 4 5 mark an bn a1 b1 mark f

Definition We let H∗

I be the set of heaps in HI with a mark in their rightmost

maximal segment. Corollary The map f induces a bijection between marked PPP and H∗

I .

Riccardo Biagioli (Universit´ e Lyon 1) joint work with M. Bousquet-M´ elou, F. Jouhet, P. Nadeau Heaps, periodic parallelogram polyominoes, and 321-avoiding affine permutations

FC elements 321-avoiding elements Parellolograms polyominoes Generating functions

From PPP to heaps of segments

Affine case

1 2 3 4 5 a1 b1 an bn f width height

width of P = number of segments in f (P) height of P = the sum of the length of the segments of f (P) area of P = the sum of right endpoints of the segments.

Riccardo Biagioli (Universit´ e Lyon 1) joint work with M. Bousquet-M´ elou, F. Jouhet, P. Nadeau Heaps, periodic parallelogram polyominoes, and 321-avoiding affine permutations

FC elements 321-avoiding elements Parellolograms polyominoes Generating functions

Generating functions for heaps of segments

Set H(x, y, q) =

• H∈H

v(H), with v(H) = xℓ(H)y|H|qe(H). where ℓ(H) is the sum of the lengths of all the segments of H; |H| is the number of segments of H; e(H) is the sum of all right endpoints of the segments of H. Proposition (Viennot (1985)) H(x, y, q) = 1 T(x, y, q) and SP(x, y, q) = T c(x, y, q) T(x, y, q) where T (resp. T c) is the signed GF for trivial heaps (resp. not touching abscissa 1), i.e. T =

T∈T (−1)|T|v(T), where T is the set of trivial heaps.

Riccardo Biagioli (Universit´ e Lyon 1) joint work with M. Bousquet-M´ elou, F. Jouhet, P. Nadeau Heaps, periodic parallelogram polyominoes, and 321-avoiding affine permutations

FC elements 321-avoiding elements Parellolograms polyominoes Generating functions

Generating functions for heaps of segments

Set H(x, y, q) =

• H∈H

v(H), with v(H) = xℓ(H)y|H|qe(H). where ℓ(H) is the sum of the lengths of all the segments of H; |H| is the number of segments of H; e(H) is the sum of all right endpoints of the segments of H. Proposition (Viennot (1985)) H(x, y, q) = 1 T(x, y, q) and SP(x, y, q) = T c(x, y, q) T(x, y, q) where T (resp. T c) is the signed GF for trivial heaps (resp. not touching abscissa 1), i.e. T =

T∈T (−1)|T|v(T), where T is the set of trivial heaps.

Riccardo Biagioli (Universit´ e Lyon 1) joint work with M. Bousquet-M´ elou, F. Jouhet, P. Nadeau Heaps, periodic parallelogram polyominoes, and 321-avoiding affine permutations

FC elements 321-avoiding elements Parellolograms polyominoes Generating functions

Generating functions for heaps of segments

Lemma (Bousquet-M´ elou, Viennot (1992)) T =

• n≥0

(−xy)nq(n+1

2 )

(q)n(xq)n and T c =

• n≥1

(−xy)nq(n+1

2 )

(q)n−1(xq)n . Since 321-avoiding permutations are in bijection with semi-pyramids, this gives back the result of Barcucci et al by setting y = 1/q (recall that we added a box in each column !) A(x) = 1 1 − xq × J(xq) J(x) , where J(x) :=

• n≥0

(−x)nq(n

2)

(q)n(xq)n

Riccardo Biagioli (Universit´ e Lyon 1) joint work with M. Bousquet-M´ elou, F. Jouhet, P. Nadeau Heaps, periodic parallelogram polyominoes, and 321-avoiding affine permutations

FC elements 321-avoiding elements Parellolograms polyominoes Generating functions

Generating functions for heaps of segments

Lemma (Bousquet-M´ elou, Viennot (1992)) T =

• n≥0

(−xy)nq(n+1

2 )

(q)n(xq)n and T c =

• n≥1

(−xy)nq(n+1

2 )

(q)n−1(xq)n . Since 321-avoiding permutations are in bijection with semi-pyramids, this gives back the result of Barcucci et al by setting y = 1/q (recall that we added a box in each column !) A(x) = 1 1 − xq × J(xq) J(x) , where J(x) :=

• n≥0

(−x)nq(n

2)

(q)n(xq)n

Riccardo Biagioli (Universit´ e Lyon 1) joint work with M. Bousquet-M´ elou, F. Jouhet, P. Nadeau Heaps, periodic parallelogram polyominoes, and 321-avoiding affine permutations

FC elements 321-avoiding elements Parellolograms polyominoes Generating functions

Adaptation to our special heaps of segments in HI

1 2 3 4 5 6 7 1 2 3 4 5 6 7 (−1)2 Map φ

1 2 3 4 5 6 7

The idea is to study the pairs (F, U) belonging to φ(HI × T ), where φ(E, T) = (E ∗ T, T). We have that Lemma The pair (F, U) belongs to φ(HI × T ) if and only if U ⊆ min(F) and one of the following cases occurs:

1 F ∈ T and |F \ U| = 1. 2 F ∈ T , F \ U1(F) ∈ HI and U1(F) ⊆ U ⊆ U2(F). Riccardo Biagioli (Universit´ e Lyon 1) joint work with M. Bousquet-M´ elou, F. Jouhet, P. Nadeau Heaps, periodic parallelogram polyominoes, and 321-avoiding affine permutations

FC elements 321-avoiding elements Parellolograms polyominoes Generating functions

Adaptation to our special heaps of segments in HI

We consider the product

• H∈HI

v(H) ×

• T∈T

(−1)|T|v(T) =

• (F,U)∈φ(HI ×T )

v(F)(−1)|U|. By the previous lemma it is equal to

• F∈T

|F|(−1)|F|−1v(F) +

• F,F\U1(F)∈HI

v(F) ×

• U1(F)⊆U⊆U2(F)

(−1)|U|. The first sum is −y∂yT while the second this is always 0. Recall: T =

• T∈T

(−1)|T|v(T) and v(H) = xℓ(H)y|H|qe(H).

Riccardo Biagioli (Universit´ e Lyon 1) joint work with M. Bousquet-M´ elou, F. Jouhet, P. Nadeau Heaps, periodic parallelogram polyominoes, and 321-avoiding affine permutations

FC elements 321-avoiding elements Parellolograms polyominoes Generating functions

Adaptation to our special heaps of segments in H∗

I

We consider the product

• H∈HI∗

v(H) ×

• T∈T

(−1)|T|v(T) =

• (F,U)∈φ(H∗

I ×T )

v(F)(−1)|U|. By the previous lemma it is equal to (we can mark each piece of F)

• F∈T

ℓ(F)(−1)|F|−1v(F) +

• F\U1(F)∈H∗

I

v(F) ×

• U1(F)⊆U⊆U2(F)

(−1)|U|. The first sum is −x∂xT while the second this is always 0. Recall: T =

• T∈T

(−1)|T|v(T) and v(H) = xℓ(H)y|H|qe(H).

Riccardo Biagioli (Universit´ e Lyon 1) joint work with M. Bousquet-M´ elou, F. Jouhet, P. Nadeau Heaps, periodic parallelogram polyominoes, and 321-avoiding affine permutations

FC elements 321-avoiding elements Parellolograms polyominoes Generating functions

Resuming

Theorem (B, Bousquet-M´ elou, Jouhet, Nadeau (2015)) The GF for periodic parallelogram polyominoes and marked ones, weighted by xhalf −perimetery#columnsqarea are respectively given by −y ∂yT T and −x ∂xT T Since marked PPP (minus those of rectangular shape) are in bijection with 321-avoiding affine permutations, we obtain (after taking care about the weight, y = 1/q) that

• A(x) = −x J′(x)

J(x) −

• n≥1

xnqn 1 − qn

Riccardo Biagioli (Universit´ e Lyon 1) joint work with M. Bousquet-M´ elou, F. Jouhet, P. Nadeau Heaps, periodic parallelogram polyominoes, and 321-avoiding affine permutations

FC elements 321-avoiding elements Parellolograms polyominoes Generating functions

Resuming

Theorem (B, Bousquet-M´ elou, Jouhet, Nadeau (2015)) The GF for periodic parallelogram polyominoes and marked ones, weighted by xhalf −perimetery#columnsqarea are respectively given by −y ∂yT T and −x ∂xT T Since marked PPP (minus those of rectangular shape) are in bijection with 321-avoiding affine permutations, we obtain (after taking care about the weight, y = 1/q) that

• A(x) = −x J′(x)

J(x) −

• n≥1

xnqn 1 − qn

Riccardo Biagioli (Universit´ e Lyon 1) joint work with M. Bousquet-M´ elou, F. Jouhet, P. Nadeau Heaps, periodic parallelogram polyominoes, and 321-avoiding affine permutations

FC elements 321-avoiding elements Parellolograms polyominoes Generating functions

Open problem

Using the bijection φ we find

• H∈Π

v(H) = −y ∂yT T =

• H∈HI

v(H) A bijection between the sets HI and the set of pyramids Π would be nice, as would be a direct way of encoding periodic parallelogram polyominoes as pyramids.

Riccardo Biagioli (Universit´ e Lyon 1) joint work with M. Bousquet-M´ elou, F. Jouhet, P. Nadeau Heaps, periodic parallelogram polyominoes, and 321-avoiding affine permutations

FC elements 321-avoiding elements Parellolograms polyominoes Generating functions

End of the talk

The end

Riccardo Biagioli (Universit´ e Lyon 1) joint work with M. Bousquet-M´ elou, F. Jouhet, P. Nadeau Heaps, periodic parallelogram polyominoes, and 321-avoiding affine permutations