Rook and Wilf equivalence of integer partitions Jonathan S. Bloom - - PowerPoint PPT Presentation

Rook and Wilf equivalence of integer partitions

Jonathan S. Bloom

Lafayette College (Joint work with Dan Saracino & Nathan McNew)

Permutation Patterns – Dartmouth College July, 2018

Some basic definitions

Some basic definitions

A partition of n is a weakly decreasing sequence λ λ1 ≥ λ2 ≥ . . . ≥ λh > 0 with |λ| = λ1 + · · · + λh = n.

Some basic definitions

A partition of n is a weakly decreasing sequence λ λ1 ≥ λ2 ≥ . . . ≥ λh > 0 with |λ| = λ1 + · · · + λh = n. Let ◮ Pn = set of all partitions of n

Some basic definitions

A partition of n is a weakly decreasing sequence λ λ1 ≥ λ2 ≥ . . . ≥ λh > 0 with |λ| = λ1 + · · · + λh = n. Let ◮ Pn = set of all partitions of n ◮ P =

• n>0

Pn.

Some basic definitions

A partition of n is a weakly decreasing sequence λ λ1 ≥ λ2 ≥ . . . ≥ λh > 0 with |λ| = λ1 + · · · + λh = n. Let ◮ Pn = set of all partitions of n ◮ P =

• n>0

Pn.

⋆ We identify partitions with Ferrers diagrams. E.g.,

4+3+2+1 = 3+3+3 = 1+1+1+1 = 4 =

Rook Theory

Rook Theory

Question: How many configurations of k non-attacking rooks can be placed on a partition (Ferrers diagram)?

R R R R R R

⋆ rooks “attack” along rows/columns.

Rook Theory

Question: How many configurations of k non-attacking rooks can be placed on a partition (Ferrers diagram)?

R R R R R R

⋆ rooks “attack” along rows/columns.

Definition

For any partition µ ∈ P we define its rook polynomial to be Rµ(q) =

• k≥0

r(µ, k)qk where r(µ, k) = number of k-configurations on µ.

Rook Theory

Question: How many configurations of k non-attacking rooks can be placed on a partition (Ferrers diagram)?

R R R R R R

⋆ rooks “attack” along rows/columns.

Definition

For any partition µ ∈ P we define its rook polynomial to be Rµ(q) =

• k≥0

r(µ, k)qk where r(µ, k) = number of k-configurations on µ. For example: ◮ R(4,2)(q) = 1 +

Rook Theory

Question: How many configurations of k non-attacking rooks can be placed on a partition (Ferrers diagram)?

R R R R R R

⋆ rooks “attack” along rows/columns.

Definition

For any partition µ ∈ P we define its rook polynomial to be Rµ(q) =

• k≥0

r(µ, k)qk where r(µ, k) = number of k-configurations on µ. For example: ◮ R(4,2)(q) = 1 + 6q +

Rook Theory

Question: How many configurations of k non-attacking rooks can be placed on a partition (Ferrers diagram)?

R R R R R R

⋆ rooks “attack” along rows/columns.

Definition

For any partition µ ∈ P we define its rook polynomial to be Rµ(q) =

• k≥0

r(µ, k)qk where r(µ, k) = number of k-configurations on µ. For example: ◮ R(4,2)(q) = 1 + 6q + 6q2

Rook Theory

Rook Theory

Definition

Two partitions µ, τ ∈ P are rook equivalent provided that Rµ(q) = Rτ(q)

⋆ µ, τ admit the same number of k-configurations.

Rook Theory

Definition

Two partitions µ, τ ∈ P are rook equivalent provided that Rµ(q) = Rτ(q)

⋆ µ, τ admit the same number of k-configurations.

◮ Rook classes for n = 6: 1 + 6q + 6q2

Rook Theory

Definition

Two partitions µ, τ ∈ P are rook equivalent provided that Rµ(q) = Rτ(q)

⋆ µ, τ admit the same number of k-configurations.

◮ Rook classes for n = 6: 1 + 6q + 6q2 1 + 6q + 4q2 1 + 6q 1 + 6q + 7q2 + q3

Rook Theory

Theorem (Foata & Sh¨ utzenberger - 1970)

Fix µ, τ ∈ Pn. The following are equivalent

Rook Theory

Theorem (Foata & Sh¨ utzenberger - 1970)

Fix µ, τ ∈ Pn. The following are equivalent (i) µ and τ are rook equivalent

Rook Theory

Theorem (Foata & Sh¨ utzenberger - 1970)

Fix µ, τ ∈ Pn. The following are equivalent (i) µ and τ are rook equivalent (ii) We have equality of the “L-multisets” {1 + µ1, 2 + µ2, 3 + µ3, . . .} = {1 + τ1, 2 + τ2, 3 + τ3, . . .}

Rook Theory

Theorem (Foata & Sh¨ utzenberger - 1970)

Fix µ, τ ∈ Pn. The following are equivalent (i) µ and τ are rook equivalent (ii) We have equality of the “L-multisets” {1 + µ1, 2 + µ2, 3 + µ3, . . .} = {1 + τ1, 2 + τ2, 3 + τ3, . . .} For example if µ = (4, 4, 3, 2, 1), then

Rook Theory

Theorem (Foata & Sh¨ utzenberger - 1970)

Fix µ, τ ∈ Pn. The following are equivalent (i) µ and τ are rook equivalent (ii) We have equality of the “L-multisets” {1 + µ1, 2 + µ2, 3 + µ3, . . .} = {1 + τ1, 2 + τ2, 3 + τ3, . . .} For example if µ = (4, 4, 3, 2, 1), then

Rook Theory

Theorem (Foata & Sh¨ utzenberger - 1970)

Fix µ, τ ∈ Pn. The following are equivalent (i) µ and τ are rook equivalent (ii) We have equality of the “L-multisets” {1 + µ1, 2 + µ2, 3 + µ3, . . .} = {1 + τ1, 2 + τ2, 3 + τ3, . . .} For example if µ = (4, 4, 3, 2, 1), then

Rook Theory

Theorem (Foata & Sh¨ utzenberger - 1970)

Fix µ, τ ∈ Pn. The following are equivalent (i) µ and τ are rook equivalent (ii) We have equality of the “L-multisets” {1 + µ1, 2 + µ2, 3 + µ3, . . .} = {1 + τ1, 2 + τ2, 3 + τ3, . . .} For example if µ = (4, 4, 3, 2, 1), then

Rook Theory

Theorem (Foata & Sh¨ utzenberger - 1970)

Fix µ, τ ∈ Pn. The following are equivalent (i) µ and τ are rook equivalent (ii) We have equality of the “L-multisets” {1 + µ1, 2 + µ2, 3 + µ3, . . .} = {1 + τ1, 2 + τ2, 3 + τ3, . . .} For example if µ = (4, 4, 3, 2, 1), then

Rook Theory

Theorem (Foata & Sh¨ utzenberger - 1970)

Fix µ, τ ∈ Pn. The following are equivalent (i) µ and τ are rook equivalent (ii) We have equality of the “L-multisets” {1 + µ1, 2 + µ2, 3 + µ3, . . .} = {1 + τ1, 2 + τ2, 3 + τ3, . . .} For example if µ = (4, 4, 3, 2, 1), then ◮ Hence the term L-multisets

Integer partition patterns

Integer partition patterns

α = contains µ =

Integer partition patterns

α = contains µ =

Integer partition patterns

α = contains µ =

Integer partition patterns

α = contains µ =

Definition (Bloom, Saracino)

We say α contains µ if one can delete rows and columns from α to obtain µ.

Integer partition patterns

α = contains µ =

Definition (Bloom, Saracino)

We say α contains µ if one can delete rows and columns from α to obtain µ. Containment defined by row-only deletion? = ⇒ Remmel’s bijection machine for partition identities

Integer partition patterns

α = contains µ =

Definition (Bloom, Saracino)

We say α contains µ if one can delete rows and columns from α to obtain µ. Containment defined by row-only deletion? = ⇒ Remmel’s bijection machine for partition identities

• ◮ Pn(µ) = set of partitions of n containing µ

Integer partition patterns

α = contains µ =

Definition (Bloom, Saracino)

We say α contains µ if one can delete rows and columns from α to obtain µ. Containment defined by row-only deletion? = ⇒ Remmel’s bijection machine for partition identities

• ◮ Pn(µ) = set of partitions of n containing µ

◮ Pn(µ, k) = {α ∈ Pn(µ) | α1 = µ1 + k}

Integer partition patterns

If µ = ∈ P10, then we have the following sets:

Integer partition patterns

If µ = ∈ P10, then we have the following sets: P14(µ, 2)

Integer partition patterns

If µ = ∈ P10, then we have the following sets: P14(µ, 2) and P14(µ, 3)

Integer partition patterns

For µ ∈ P define Pµ(q) =

• n≥0

|Pn(µ)|qn

Integer partition patterns

For µ ∈ P define Pµ(q) =

• n≥0

|Pn(µ)|qn and Pµ,k(q) =

• n≥0

|Pn(µ, k)|qn

Integer partition patterns

For µ ∈ P define Pµ(q) =

• n≥0

|Pn(µ)|qn and Pµ,k(q) =

• n≥0

|Pn(µ, k)|qn

Definition

We say partitions µ, τ are Wilf equivalent provided Pµ(q) = Pτ(q).

Integer partition patterns

For µ ∈ P define Pµ(q) =

• n≥0

|Pn(µ)|qn and Pµ,k(q) =

• n≥0

|Pn(µ, k)|qn

Definition

We say partitions µ, τ are Wilf equivalent provided Pµ(q) = Pτ(q). Refining this, µ, τ are width-Wilf equivalent provided Pµ,k(q) = Pτ,k(q) (for all k ≥ 0)

Integer partition patterns

Wilf classes for n = 6:

Integer partition patterns

Wilf classes for n = 6:

Integer partition patterns

Wilf classes for n = 6:

⋆

Same as rook classes!

Integer partition patterns

Wilf classes for n = 6:

⋆

Same as rook classes! Can we prove it?

Our main result

Theorem (Bloom, Saracino)

Rook equivalence ⇐ ⇒ Wilf equivalence.

Our main result

Theorem (Bloom, Saracino)

Rook equivalence ⇐ ⇒ Wilf equivalence.    Also, assuming µ1 = τ1 for µ, τ ∈ P, then rook equivalence ⇐ ⇒ width-Wilf equivalence Pµ,1(q) = Pτ,1(q) ⇐ ⇒ width-Wilf equivalent   

Our main result

Theorem (Bloom, Saracino)

Rook equivalence ⇐ ⇒ Wilf equivalence.    Also, assuming µ1 = τ1 for µ, τ ∈ P, then rook equivalence ⇐ ⇒ width-Wilf equivalence Pµ,1(q) = Pτ,1(q) ⇐ ⇒ width-Wilf equivalent    To prove... (⇐ =) if µ and ν are not rook equivalent, then there exists an injection PN(µ) → PN(ν) for a specific N.

Our main result

Theorem (Bloom, Saracino)

Rook equivalence ⇐ ⇒ Wilf equivalence.    Also, assuming µ1 = τ1 for µ, τ ∈ P, then rook equivalence ⇐ ⇒ width-Wilf equivalence Pµ,1(q) = Pτ,1(q) ⇐ ⇒ width-Wilf equivalent    To prove... (⇐ =) if µ and ν are not rook equivalent, then there exists an injection PN(µ) → PN(ν) for a specific N. (= ⇒) Characterize Pµ,k(q) in terms of “L-multisets”.

Proof sketch: rook = ⇒ Wilf

Idea: Build P(µ) by inserting rows/columns into µ.

Proof sketch: rook = ⇒ Wilf

Idea: Build P(µ) by inserting rows/columns into µ. Problem: Multiple ways to build same partition!!

Proof sketch: rook = ⇒ Wilf

Idea: Build P(µ) by inserting rows/columns into µ. Problem: Multiple ways to build same partition!!

µ

Proof sketch: rook = ⇒ Wilf

Idea: Build P(µ) by inserting rows/columns into µ. Problem: Multiple ways to build same partition!!

µ

= ⇒

Proof sketch: rook = ⇒ Wilf

Idea: Build P(µ) by inserting rows/columns into µ. Problem: Multiple ways to build same partition!!

µ

= ⇒ = ⇒

Proof sketch: rook = ⇒ Wilf

Idea: Build P(µ) by inserting rows/columns into µ. Problem: Multiple ways to build same partition!!

µ

= ⇒ = ⇒

• r

µ

Proof sketch: rook = ⇒ Wilf

Idea: Build P(µ) by inserting rows/columns into µ. Problem: Multiple ways to build same partition!!

µ

= ⇒ = ⇒

• r

µ

= ⇒

Proof sketch: rook = ⇒ Wilf

Idea: Build P(µ) by inserting rows/columns into µ. Problem: Multiple ways to build same partition!!

µ

= ⇒ = ⇒

• r

µ

= ⇒ = ⇒

Proof sketch: rook = ⇒ Wilf

Idea: Build P(µ) by inserting rows/columns into µ. Problem: Multiple ways to build same partition!!

µ

= ⇒ = ⇒

• r

µ

= ⇒ = ⇒

⋆ Need Inclusion/Exclusion!

Proof sketch: rook = ⇒ Wilf

Proof sketch: rook = ⇒ Wilf

Define the sum: (2, 2, 2, 1) + (3, 2, 2, 1) = (5, 4, 4, 2)

Proof sketch: rook = ⇒ Wilf

Define the sum: (2, 2, 2, 1) + (3, 2, 2, 1) = (5, 4, 4, 2) + :=

⋆ An operation on columns

Proof sketch: rook = ⇒ Wilf

Define the sum: (2, 2, 2, 1) + (3, 2, 2, 1) = (5, 4, 4, 2) + :=

⋆ An operation on columns

Define the join: (32, 22, 12) ∨ (3, 23, 1) = (3max (2,1), 2max (2,3), 1max (2,1))

Proof sketch: rook = ⇒ Wilf

Define the sum: (2, 2, 2, 1) + (3, 2, 2, 1) = (5, 4, 4, 2) + :=

⋆ An operation on columns

Define the join: (32, 22, 12) ∨ (3, 23, 1) = (3max (2,1), 2max (2,3), 1max (2,1)) ∨ :=

⋆ An operation on rows

Proof sketch: rook = ⇒ Wilf

Definition

For any (finite) S ⊂ P we define ∨S : P → P as ∨S(µ) :=

• α∈S

(µ + α)

Proof sketch: rook = ⇒ Wilf

Definition

For any (finite) S ⊂ P we define ∨S : P → P as ∨S(µ) :=

• α∈S

(µ + α) ◮ Using inclusion/exclusion we have Pµ,k(q) =

• S

(−1)|S|+1 q|∨S(µ)| (1 − q) · · · (1 − qk+µ1), S ranges over sets of partitions with k columns.

Proof sketch: rook = ⇒ Wilf

Definition

For any (finite) S ⊂ P we define ∨S : P → P as ∨S(µ) :=

• α∈S

(µ + α) ◮ Using inclusion/exclusion we have Pµ,k(q) =

• S

(−1)|S|+1 q|∨S(µ)| (1 − q) · · · (1 − qk+µ1), S ranges over sets of partitions with k columns. ◮ MANY terms cancel

Proof sketch: rook = ⇒ Wilf

Definition

For any (finite) S ⊂ P we define ∨S : P → P as ∨S(µ) :=

• α∈S

(µ + α) ◮ Using inclusion/exclusion we have Pµ,k(q) =

• S

(−1)|S|+1 q|∨S(µ)| (1 − q) · · · (1 − qk+µ1), S ranges over sets of partitions with k columns. ◮ MANY terms cancel

⇒ Characterize when the operators ∨S = ∨T for subsets S and T.

Proof sketch: rook = ⇒ Wilf

Definition

A (µ, k)-marked structure is a triple (µ, σ, A):

µ

Proof sketch: rook = ⇒ Wilf

Definition

A (µ, k)-marked structure is a triple (µ, σ, A):

µ

Proof sketch: rook = ⇒ Wilf

Definition

A (µ, k)-marked structure is a triple (µ, σ, A):

µ σ

Proof sketch: rook = ⇒ Wilf

Definition

A (µ, k)-marked structure is a triple (µ, σ, A):

µ σ

• marked cols in σ
• A = {1, 4, 5}

Proof sketch: rook = ⇒ Wilf

Definition

A (µ, k)-marked structure is a triple (µ, σ, A):

µ σ

• marked cols in σ
• A = {1, 4, 5}

Proof sketch: rook = ⇒ Wilf

Definition

A (µ, k)-marked structure is a triple (µ, σ, A):

µ σ

• marked cols in σ
• A = {1, 4, 5}

where σ1 = k.

Proof sketch: rook = ⇒ Wilf

Definition

A (µ, k)-marked structure is a triple (µ, σ, A):

µ σ

• marked cols in σ
• A = {1, 4, 5}

where σ1 = k. ◮ Let M(µ, k) be the set of all such (µ, σ, A)

Proof sketch: rook = ⇒ Wilf

Definition

A (µ, k)-marked structure is a triple (µ, σ, A):

µ σ

• marked cols in σ
• A = {1, 4, 5}

where σ1 = k. ◮ Let M(µ, k) be the set of all such (µ, σ, A) ◮ |(µ, σ, A)| = |µ| +

Proof sketch: rook = ⇒ Wilf

Definition

A (µ, k)-marked structure is a triple (µ, σ, A):

µ σ

• marked cols in σ
• A = {1, 4, 5}

where σ1 = k. ◮ Let M(µ, k) be the set of all such (µ, σ, A) ◮ |(µ, σ, A)| = |µ| + 6 + 4 + 2

• len. of unmarked columns

+

Proof sketch: rook = ⇒ Wilf

Definition

A (µ, k)-marked structure is a triple (µ, σ, A):

µ σ

• marked cols in σ
• A = {1, 4, 5}

where σ1 = k. ◮ Let M(µ, k) be the set of all such (µ, σ, A) ◮ |(µ, σ, A)| = |µ| + 6 + 4 + 2

• len. of unmarked columns

+

Proof sketch: rook = ⇒ Wilf

Definition

A (µ, k)-marked structure is a triple (µ, σ, A):

µ σ

• marked cols in σ
• A = {1, 4, 5}

where σ1 = k. ◮ Let M(µ, k) be the set of all such (µ, σ, A) ◮ |(µ, σ, A)| = |µ| + 6 + 4 + 2

• len. of unmarked columns

+8 +

Proof sketch: rook = ⇒ Wilf

Definition

A (µ, k)-marked structure is a triple (µ, σ, A):

µ σ

• marked cols in σ
• A = {1, 4, 5}

where σ1 = k. ◮ Let M(µ, k) be the set of all such (µ, σ, A) ◮ |(µ, σ, A)| = |µ| + 6 + 4 + 2

• len. of unmarked columns

+8 +

Proof sketch: rook = ⇒ Wilf

Definition

A (µ, k)-marked structure is a triple (µ, σ, A):

µ σ

• marked cols in σ
• A = {1, 4, 5}

where σ1 = k. ◮ Let M(µ, k) be the set of all such (µ, σ, A) ◮ |(µ, σ, A)| = |µ| + 6 + 4 + 2

• len. of unmarked columns

+8 + 10 +

Proof sketch: rook = ⇒ Wilf

Definition

A (µ, k)-marked structure is a triple (µ, σ, A):

µ σ

• marked cols in σ
• A = {1, 4, 5}

where σ1 = k. ◮ Let M(µ, k) be the set of all such (µ, σ, A) ◮ |(µ, σ, A)| = |µ| + 6 + 4 + 2

• len. of unmarked columns

+8 + 10 +

Proof sketch: rook = ⇒ Wilf

Definition

A (µ, k)-marked structure is a triple (µ, σ, A):

µ σ

• marked cols in σ
• A = {1, 4, 5}

where σ1 = k. ◮ Let M(µ, k) be the set of all such (µ, σ, A) ◮ |(µ, σ, A)| = |µ| + 6 + 4 + 2

• len. of unmarked columns

+8 + 10 + 10

Proof sketch: rook = ⇒ Wilf

Definition

A (µ, k)-marked structure is a triple (µ, σ, A):

µ σ

• marked cols in σ
• A = {1, 4, 5}

where σ1 = k. ◮ Let M(µ, k) be the set of all such (µ, σ, A) ◮ |(µ, σ, A)| = |µ| + 6 + 4 + 2

• len. of unmarked columns

+8 + 10 + 10

⋆ Marked structures “index” L-multisets!

Proof sketch: rook = ⇒ Wilf

Proof sketch: rook = ⇒ Wilf

We had Pµ,k(q) =

• n≥1

|Pn(µ, k)|qn =

• S

(−1)|S|+1 q|∨S(µ)| (1 − q) · · · (1 − qk+µ1)

Proof sketch: rook = ⇒ Wilf

We had Pµ,k(q) =

• n≥1

|Pn(µ, k)|qn =

• S

(−1)|S|+1 q|∨S(µ)| (1 − q) · · · (1 − qk+µ1)

Proof sketch: rook = ⇒ Wilf

We had Pµ,k(q) =

• n≥1

|Pn(µ, k)|qn =

• S

(−1)|S|+1 q|∨S(µ)| (1 − q) · · · (1 − qk+µ1) after loads of cancellation... Pµ,k(q) = q|µ| (1 − q) · · · (1 − qk+µ1)

• (µ,σ,A)∈M(µ,k)

(−1)|A|q|(µ,σ,A)|

Enumeration results

Enumeration results

Theorem (Bloom, McNew)

Let µ ∈ P so that |µi − µj| > 1 for i = j, then

• n≥0

|Pn \ Pn(µ)|qn is rational.

Enumeration results

Theorem (Bloom, McNew)

Let µ ∈ P so that |µi − µj| > 1 for i = j, then

• n≥0

|Pn \ Pn(µ)|qn is rational.

Enumeration results

Theorem (Bloom, McNew)

Let µ ∈ P so that |µi − µj| > 1 for i = j, then

• n≥0

|Pn \ Pn(µ)|qn is rational.

Corollary (Bloom, McNew)

Fix N ≥ 0. Then the GF for partitions τ with |τi − τk| < N is rational.

Enumeration results

Theorem (Bloom, McNew)

Let µ ∈ P so that |µi − µj| > 1 for i = j, then

• n≥0

|Pn \ Pn(µ)|qn is rational.

Corollary (Bloom, McNew)

Fix N ≥ 0. Then the GF for partitions τ with |τi − τk| < N is rational. = ⇒ τ avoids µ = (N + 1, 1)

Thank You!