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 λ 1 ≥ λ 2 ≥ . . . ≥ λ h > 0 with | λ | = λ 1 + · · · + λ h = n .

Some basic definitions A partition of n is a weakly decreasing sequence λ λ 1 ≥ λ 2 ≥ . . . ≥ λ h > 0 with | λ | = λ 1 + · · · + λ h = n . Let ◮ P n = set of all partitions of n

Some basic definitions A partition of n is a weakly decreasing sequence λ λ 1 ≥ λ 2 ≥ . . . ≥ λ h > 0 with | λ | = λ 1 + · · · + λ h = n . Let ◮ P n = set of all partitions of n � ◮ P = P n . n > 0

Some basic definitions A partition of n is a weakly decreasing sequence λ λ 1 ≥ λ 2 ≥ . . . ≥ λ h > 0 with | λ | = λ 1 + · · · + λ h = n . Let ◮ P n = set of all partitions of n � ◮ P = P n . n > 0 ⋆ 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 � r ( µ, k ) q k R µ ( q ) = k ≥ 0 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 � r ( µ, k ) q k R µ ( q ) = k ≥ 0 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 � r ( µ, k ) q k R µ ( q ) = k ≥ 0 where r ( µ, k ) = number of k -configurations on µ . For example: ◮ R (4 , 2) ( q ) = 1 + 6 q +

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 � r ( µ, k ) q k R µ ( q ) = k ≥ 0 where r ( µ, k ) = number of k -configurations on µ . For example: ◮ R (4 , 2) ( q ) = 1 + 6 q + 6 q 2

Rook Theory

Rook Theory Definition Two partitions µ, τ ∈ P are rook equivalent provided that R µ ( q ) = R τ ( q ) ⋆ µ, τ admit the same number of k -configurations.

Rook Theory Definition Two partitions µ, τ ∈ P are rook equivalent provided that R µ ( q ) = R τ ( q ) ⋆ µ, τ admit the same number of k -configurations. ◮ Rook classes for n = 6: 1 + 6 q + 6 q 2

Rook Theory Definition Two partitions µ, τ ∈ P are rook equivalent provided that R µ ( q ) = R τ ( q ) ⋆ µ, τ admit the same number of k -configurations. ◮ Rook classes for n = 6: 1 + 6 q + 6 q 2 1 + 6 q + 7 q 2 + q 3 1 + 6 q + 4 q 2 1 + 6 q

Rook Theory Theorem (Foata & Sh¨ utzenberger - 1970) Fix µ, τ ∈ P n . The following are equivalent

Rook Theory Theorem (Foata & Sh¨ utzenberger - 1970) Fix µ, τ ∈ P n . The following are equivalent (i) µ and τ are rook equivalent

Rook Theory Theorem (Foata & Sh¨ utzenberger - 1970) Fix µ, τ ∈ P n . The following are equivalent (i) µ and τ are rook equivalent (ii) We have equality of the “L-multisets” { 1 + µ 1 , 2 + µ 2 , 3 + µ 3 , . . . } = { 1 + τ 1 , 2 + τ 2 , 3 + τ 3 , . . . }

Rook Theory Theorem (Foata & Sh¨ utzenberger - 1970) Fix µ, τ ∈ P n . The following are equivalent (i) µ and τ are rook equivalent (ii) We have equality of the “L-multisets” { 1 + µ 1 , 2 + µ 2 , 3 + µ 3 , . . . } = { 1 + τ 1 , 2 + τ 2 , 3 + τ 3 , . . . } For example if µ = (4 , 4 , 3 , 2 , 1), then

Rook Theory Theorem (Foata & Sh¨ utzenberger - 1970) Fix µ, τ ∈ P n . The following are equivalent (i) µ and τ are rook equivalent (ii) We have equality of the “L-multisets” { 1 + µ 1 , 2 + µ 2 , 3 + µ 3 , . . . } = { 1 + τ 1 , 2 + τ 2 , 3 + τ 3 , . . . } For example if µ = (4 , 4 , 3 , 2 , 1), then

Rook Theory Theorem (Foata & Sh¨ utzenberger - 1970) Fix µ, τ ∈ P n . The following are equivalent (i) µ and τ are rook equivalent (ii) We have equality of the “L-multisets” { 1 + µ 1 , 2 + µ 2 , 3 + µ 3 , . . . } = { 1 + τ 1 , 2 + τ 2 , 3 + τ 3 , . . . } For example if µ = (4 , 4 , 3 , 2 , 1), then

Rook Theory Theorem (Foata & Sh¨ utzenberger - 1970) Fix µ, τ ∈ P n . The following are equivalent (i) µ and τ are rook equivalent (ii) We have equality of the “L-multisets” { 1 + µ 1 , 2 + µ 2 , 3 + µ 3 , . . . } = { 1 + τ 1 , 2 + τ 2 , 3 + τ 3 , . . . } For example if µ = (4 , 4 , 3 , 2 , 1), then

Rook Theory Theorem (Foata & Sh¨ utzenberger - 1970) Fix µ, τ ∈ P n . The following are equivalent (i) µ and τ are rook equivalent (ii) We have equality of the “L-multisets” { 1 + µ 1 , 2 + µ 2 , 3 + µ 3 , . . . } = { 1 + τ 1 , 2 + τ 2 , 3 + τ 3 , . . . } For example if µ = (4 , 4 , 3 , 2 , 1), then

Rook Theory Theorem (Foata & Sh¨ utzenberger - 1970) Fix µ, τ ∈ P n . The following are equivalent (i) µ and τ are rook equivalent (ii) We have equality of the “L-multisets” { 1 + µ 1 , 2 + µ 2 , 3 + µ 3 , . . . } = { 1 + τ 1 , 2 + τ 2 , 3 + τ 3 , . . . } 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 ◮ P n ( 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 ◮ P n ( µ ) = set of partitions of n containing µ ◮ P n ( µ, k ) = { α ∈ P n ( µ ) | α 1 = µ 1 + k }

Integer partition patterns If µ = ∈ P 10 , then we have the following sets:

Integer partition patterns If µ = ∈ P 10 , then we have the following sets: P 14 ( µ, 2)

Integer partition patterns If µ = ∈ P 10 , then we have the following sets: P 14 ( µ, 2) and P 14 ( µ, 3)

Integer partition patterns For µ ∈ P define � | P n ( µ ) | q n P µ ( q ) = n ≥ 0

Integer partition patterns For µ ∈ P define � � | P n ( µ ) | q n | P n ( µ, k ) | q n P µ ( q ) = and P µ, k ( q ) = n ≥ 0 n ≥ 0

Integer partition patterns For µ ∈ P define � � | P n ( µ ) | q n | P n ( µ, k ) | q n P µ ( q ) = and P µ, k ( q ) = n ≥ 0 n ≥ 0 Definition We say partitions µ, τ are Wilf equivalent provided P µ ( q ) = P τ ( q ) .

Integer partition patterns For µ ∈ P define � � | P n ( µ ) | q n | P n ( µ, k ) | q n P µ ( q ) = and P µ, k ( q ) = n ≥ 0 n ≥ 0 Definition We say partitions µ, τ are Wilf equivalent provided P µ ( q ) = P τ ( q ) . Refining this, µ, τ are width-Wilf equivalent provided P µ, k ( q ) = P τ, 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 µ 1 = τ 1 for µ, τ ∈ P , then   rook equivalence ⇐ ⇒ width-Wilf equivalence   P µ, 1 ( q ) = P τ, 1 ( q ) ⇐ ⇒ width-Wilf equivalent