Extension of Andrews’ identity

Theorem (Andrews (2019))

LetOd(n)denote the number of partitions of n in which the odd parts are distinct and each positive odd integer smaller than the largest odd part must appear as a part. Then

p^od_eu(n) =Od(n),

where p^od_eu(n)denotes the number of partitions of n in which each even part is less than each odd part and odd parts are distinct.

Ex: The 6 partitions enumerated byOd(9) are 8 + 1, 6 + 2 + 1, 5 + 3 + 1, 4 + 4 + 1, 4 + 2 + 2 + 1, 2 + 2 + 2 + 2 + 1 and those enumerated by p_eu^od(9) are 9, 7 + 2, 5 + 4, 5 + 3 + 1, 5 + 2 + 2, 3 + 2 + 2 + 2.

Extension of Andrews’ identity

Theorem (Andrews (2019))

LetOd(n)denote the number of partitions of n in which the odd parts are distinct and each positive odd integer smaller than the largest odd part must appear as a part. Then

p^od_eu(n) =Od(n),

where p^od_eu(n)denotes the number of partitions of n in which each even part is less than each odd part and odd parts are distinct.

Ex: The 6 partitions enumerated byOd(9) are 8 + 1, 6 + 2 + 1, 5 + 3 + 1, 4 + 4 + 1, 4 + 2 + 2 + 1, 2 + 2 + 2 + 2 + 1 and those enumerated by p_eu^od(9) are 9, 7 + 2, 5 + 4, 5 + 3 + 1, 5 + 2 + 2, 3 + 2 + 2 + 2.

Koustav Banerjee BSP and Partition identities 33 / 45

Extension of Andrews’ identity

Definition 1:

P_eu^ou(n) :=

(

λ`n: (1) all the odd parts ofλare unrestricted, (2) each even part ofλis less than each odd part ofλ

) ,

p_eu^ou(n) := #{λ`n:λ∈P_eu^ou(n)}. For example,p_eu^ou(9) = 12

(9,7+2,7+1+1,5+4,5+3+1,5+2+2,5+1+1+1+1,3+3+3,3+3+ 1+1+1,3+2+2+2,3+1+1+1+1+1+1,1+1+1+1+1+1+1+1+1).

Extension of Andrews’ identity

Definition 1:

P_eu^ou(n) :=

(

λ`n: (1) all the odd parts ofλare unrestricted, (2) each even part ofλis less than each odd part ofλ

) ,

p_eu^ou(n) := #{λ`n:λ∈P_eu^ou(n)}.

For example,p_eu^ou(9) = 12

(9,7+2,7+1+1,5+4,5+3+1,5+2+2,5+1+1+1+1,3+3+3,3+3+ 1+1+1,3+2+2+2,3+1+1+1+1+1+1,1+1+1+1+1+1+1+1+1).

Koustav Banerjee BSP and Partition identities 34 / 45

Extension of Andrews’ identity

Definition 1:

P_eu^ou(n) :=

(

λ`n: (1) all the odd parts ofλare unrestricted, (2) each even part ofλis less than each odd part ofλ

) ,

p_eu^ou(n) := #{λ`n:λ∈P_eu^ou(n)}.

For example,p_eu^ou(9) = 12

(9,7+2,7+1+1,5+4,5+3+1,5+2+2,5+1+1+1+1,3+3+3,3+3+

1+1+1,3+2+2+2,3+1+1+1+1+1+1,1+1+1+1+1+1+1+1+1).

Extension of Andrews’ identity

Definition 2: For λ`nsuch that an odd integer must appear as a part ofλ,

OMax(λ) := greatest odd part of λ, EMax(λ) :=

greatest even part of λ, if even parts occur inλ,

0, otherwise

Koustav Banerjee BSP and Partition identities 35 / 45

Extension of Andrews’ identity

Definition 2: For λ`nsuch that an odd integer must appear as a part ofλ,

OMax(λ) := greatest odd part of λ,

EMax(λ) :=

greatest even part of λ, if even parts occur inλ,

0, otherwise

Extension of Andrews’ identity

Definition 2: For λ`nsuch that an odd integer must appear as a part ofλ,

OMax(λ) := greatest odd part of λ, EMax(λ) :=

greatest even part of λ, if even parts occur inλ,

0, otherwise

Koustav Banerjee BSP and Partition identities 35 / 45

Extension of Andrews’ identity

Definition 2: For λ`nsuch that an odd integer must appear as a part ofλ,

OMax(λ) := greatest odd part of λ, EMax(λ) :=

greatest even part of λ, if even parts occur inλ,

0, otherwise

Extension of Andrews’ identity

Definition 2: For λ`nsuch that an odd integer must appear as a part ofλ,

OMax(λ) := greatest odd part of λ, EMax(λ) :=

greatest even part of λ, if even parts occur inλ,

0, otherwise

Koustav Banerjee BSP and Partition identities 35 / 45

Extension of Andrews’ identity

Definition 2: For λ`nsuch that an odd integer must appear as a part ofλ,

OMax(λ) := greatest odd part of λ, EMax(λ) :=

greatest even part of λ, if even parts occur inλ,

0, otherwise

Extension of Andrews’ identity

Definition 4:

OEMaxDiff^∗(λ) = min{OEMaxDiff (λ⁰) :λ⁰ ∈O_u(n)}.

O_u^∗(n) :={λ∈Ou(n) : OEMaxDiff^∗(λ)}. For example,o_u^∗(9) = 12

(8 + 1,6 + 2 + 1,5 + 3 + 1,4 + 4 + 1,4 + 3 + 1 + 1,4 + 2 + 2 + 1,3 + 2 + 1 + 1 + 1 + 1,2 + 2 + 2 + 2 + 1,3 + 3 + 1 + 1 + 1,3 + 1 + 1 + 1 + 1 + 1 + 1,2 + 2 + 1 + 1 + 1 + 1 + 1,1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1). According to our definition, the partition λ= (6,1,1,1)∈/O_u^∗(9) but the partition (4,3,1,1)∈O_u^∗(9).

Theorem (B., Dastidar (2019)) o_u^∗(n) =p_eu^ou(n)

Koustav Banerjee BSP and Partition identities 36 / 45

Extension of Andrews’ identity

Definition 4:

OEMaxDiff^∗(λ) = min{OEMaxDiff (λ⁰) :λ⁰ ∈O_u(n)}.

O_u^∗(n) :={λ∈Ou(n) : OEMaxDiff^∗(λ)}.

For example,o_u^∗(9) = 12

(8 + 1,6 + 2 + 1,5 + 3 + 1,4 + 4 + 1,4 + 3 + 1 + 1,4 + 2 + 2 + 1,3 + 2 + 1 + 1 + 1 + 1,2 + 2 + 2 + 2 + 1,3 + 3 + 1 + 1 + 1,3 + 1 + 1 + 1 + 1 + 1 + 1,2 + 2 + 1 + 1 + 1 + 1 + 1,1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1). According to our definition, the partition λ= (6,1,1,1)∈/O_u^∗(9) but the partition (4,3,1,1)∈O_u^∗(9).

Theorem (B., Dastidar (2019)) o_u^∗(n) =p_eu^ou(n)

Extension of Andrews’ identity

Definition 4:

OEMaxDiff^∗(λ) = min{OEMaxDiff (λ⁰) :λ⁰ ∈O_u(n)}.

O_u^∗(n) :={λ∈Ou(n) : OEMaxDiff^∗(λ)}.

For example,o_u^∗(9) = 12

(8 + 1,6 + 2 + 1,5 + 3 + 1,4 + 4 + 1,4 + 3 + 1 + 1,4 + 2 + 2 + 1,3 + 2 + 1 + 1 + 1 + 1,2 + 2 + 2 + 2 + 1,3 + 3 + 1 + 1 + 1,3 + 1 + 1 + 1 + 1 + 1 + 1,2 + 2 + 1 + 1 + 1 + 1 + 1,1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1).

According to our definition, the partition λ= (6,1,1,1)∈/O_u^∗(9) but the partition (4,3,1,1)∈O_u^∗(9).

Theorem (B., Dastidar (2019)) o_u^∗(n) =p_eu^ou(n)

Koustav Banerjee BSP and Partition identities 36 / 45

Extension of Andrews’ identity

Definition 4:

OEMaxDiff^∗(λ) = min{OEMaxDiff (λ⁰) :λ⁰ ∈O_u(n)}.

O_u^∗(n) :={λ∈Ou(n) : OEMaxDiff^∗(λ)}.

For example,o_u^∗(9) = 12

(8 + 1,6 + 2 + 1,5 + 3 + 1,4 + 4 + 1,4 + 3 + 1 + 1,4 + 2 + 2 + 1,3 + 2 + 1 + 1 + 1 + 1,2 + 2 + 2 + 2 + 1,3 + 3 + 1 + 1 + 1,3 + 1 + 1 + 1 + 1 + 1 + 1,2 + 2 + 1 + 1 + 1 + 1 + 1,1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1).

According to our definition, the partition λ= (6,1,1,1)∈/O_u^∗(9) but the partition (4,3,1,1)∈O_u^∗(9).

Theorem (B., Dastidar (2019)) o_u^∗(n) =p_eu^ou(n)

Extension of Andrews’ identity

Definition 4:

OEMaxDiff^∗(λ) = min{OEMaxDiff (λ⁰) :λ⁰ ∈O_u(n)}.

O_u^∗(n) :={λ∈Ou(n) : OEMaxDiff^∗(λ)}.

For example,o_u^∗(9) = 12

(8 + 1,6 + 2 + 1,5 + 3 + 1,4 + 4 + 1,4 + 3 + 1 + 1,4 + 2 + 2 + 1,3 + 2 + 1 + 1 + 1 + 1,2 + 2 + 2 + 2 + 1,3 + 3 + 1 + 1 + 1,3 + 1 + 1 + 1 + 1 + 1 + 1,2 + 2 + 1 + 1 + 1 + 1 + 1,1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1).

According to our definition, the partition λ= (6,1,1,1)∈/O_u^∗(9) but the partition (4,3,1,1)∈O_u^∗(9).

Theorem (B., Dastidar (2019)) o_u^∗(n) =p_eu^ou(n)

Koustav Banerjee BSP and Partition identities 36 / 45

Extension of Andrews’ identity

Proof Sketch: (O_u^∗(n)−→P_eu^ou(n))

First, consider the Young diagramYλ for the partition λ= (λ1, λ2, . . . , λ`)∈O_u^∗(n).

We separateλintoλ⁰ = (λo₁, λo₂, . . . , λo_r) where 1≤oi≤`and λ⁰⁰= (λe₁, λe₂, . . . , λe_t) where 1≤oj≤l according to the odd and even parts, respectively with corresponding Young diagrams Y_λ⁰ and Y_λ⁰⁰.

Next, we joinY_λ⁰ andY_λ⁰⁰ by successively adjoining their rows with respect to the ordering of the parts inλ⁰, λ⁰⁰, respectively, starting with the largest one and end with the smallest one with restricting Young diagram, say,Y_λ⁰⁰⁰.

Now, we consider the following three cases:

(1) If the number of odd parts is equal to the number of even parts in a partition λ∈O_u^∗(n), then Y_λ⁰⁰⁰ is withλ⁰⁰⁰∈P_eu^ou(n) as for λ⁰ = (λo₁, . . . , λo_r) andλ⁰⁰= (λe₁, . . . , λe_r), the resulting partition λ⁰⁰⁰ = (λo1+λe1, . . . , λor+λer).

Extension of Andrews’ identity

Proof Sketch: (O_u^∗(n)−→P_eu^ou(n))

First, consider the Young diagramYλ for the partition λ= (λ1, λ2, . . . , λ`)∈O_u^∗(n).

We separateλintoλ⁰ = (λo₁, λo₂, . . . , λo_r) where 1≤oi≤`and λ⁰⁰= (λe₁, λe₂, . . . , λe_t) where 1≤oj≤l according to the odd and even parts, respectively with corresponding Young diagrams Y_λ⁰ and Y_λ⁰⁰.

Next, we joinY_λ⁰ andY_λ⁰⁰ by successively adjoining their rows with respect to the ordering of the parts inλ⁰, λ⁰⁰, respectively, starting with the largest one and end with the smallest one with restricting Young diagram, say,Y_λ⁰⁰⁰.

Now, we consider the following three cases:

(1) If the number of odd parts is equal to the number of even parts in a partition λ∈O_u^∗(n), then Y_λ⁰⁰⁰ is withλ⁰⁰⁰∈P_eu^ou(n) as for λ⁰ = (λo₁, . . . , λo_r) andλ⁰⁰= (λe₁, . . . , λe_r), the resulting partition λ⁰⁰⁰ = (λo1+λe1, . . . , λor+λer).

Koustav Banerjee BSP and Partition identities 37 / 45

Extension of Andrews’ identity

Proof Sketch: (O_u^∗(n)−→P_eu^ou(n))

First, consider the Young diagramYλ for the partition λ= (λ1, λ2, . . . , λ`)∈O_u^∗(n).

We separateλintoλ⁰ = (λo₁, λo₂, . . . , λo_r) where 1≤oi≤`and λ⁰⁰= (λe₁, λe₂, . . . , λe_t) where 1≤oj≤l according to the odd and even parts, respectively with corresponding Young diagrams Y_λ⁰ and Y_λ⁰⁰.

Next, we joinY_λ⁰ andY_λ⁰⁰ by successively adjoining their rows with respect to the ordering of the parts inλ⁰, λ⁰⁰, respectively, starting with the largest one and end with the smallest one with restricting Young diagram, say,Y_λ⁰⁰⁰.

Now, we consider the following three cases:

(1) If the number of odd parts is equal to the number of even parts in a partition λ∈O_u^∗(n), then Y_λ⁰⁰⁰ is withλ⁰⁰⁰∈P_eu^ou(n) as for λ⁰ = (λo₁, . . . , λo_r) andλ⁰⁰= (λe₁, . . . , λe_r), the resulting partition λ⁰⁰⁰ = (λo1+λe1, . . . , λor+λer).

Extension of Andrews’ identity

Proof Sketch: (O_u^∗(n)−→P_eu^ou(n))

First, consider the Young diagramYλ for the partition λ= (λ1, λ2, . . . , λ`)∈O_u^∗(n).

We separateλintoλ⁰ = (λo₁, λo₂, . . . , λo_r) where 1≤oi≤`and λ⁰⁰= (λe₁, λe₂, . . . , λe_t) where 1≤oj≤l according to the odd and even parts, respectively with corresponding Young diagrams Y_λ⁰ and Y_λ⁰⁰.

Next, we joinY_λ⁰ andY_λ⁰⁰ by successively adjoining their rows with respect to the ordering of the parts inλ⁰, λ⁰⁰, respectively, starting with the largest one and end with the smallest one with restricting Young diagram, say,Y_λ⁰⁰⁰.

Now, we consider the following three cases:

(1) If the number of odd parts is equal to the number of even parts in a partition λ∈O_u^∗(n), then Y_λ⁰⁰⁰ is withλ⁰⁰⁰∈P_eu^ou(n) as for λ⁰ = (λo₁, . . . , λo_r) andλ⁰⁰= (λe₁, . . . , λe_r), the resulting partition λ⁰⁰⁰ = (λo1+λe1, . . . , λor+λer).

Koustav Banerjee BSP and Partition identities 37 / 45

Extension of Andrews’ identity

Proof Sketch: (O_u^∗(n)−→P_eu^ou(n))

First, consider the Young diagramYλ for the partition λ= (λ1, λ2, . . . , λ`)∈O_u^∗(n).

We separateλintoλ⁰ = (λo₁, λo₂, . . . , λo_r) where 1≤oi≤`and λ⁰⁰= (λe₁, λe₂, . . . , λe_t) where 1≤oj≤l according to the odd and even parts, respectively with corresponding Young diagrams Y_λ⁰ and Y_λ⁰⁰.

Next, we joinY_λ⁰ andY_λ⁰⁰ by successively adjoining their rows with respect to the ordering of the parts inλ⁰, λ⁰⁰, respectively, starting with the largest one and end with the smallest one with restricting Young diagram, say,Y_λ⁰⁰⁰.

Now, we consider the following three cases:

(1) If the number of odd parts is equal to the number of even parts in a partition λ∈O_u^∗(n), then Y_λ⁰⁰⁰ is withλ⁰⁰⁰∈P_eu^ou(n) as for λ⁰ = (λo₁, . . . , λo_r) andλ⁰⁰= (λe₁, . . . , λe_r), the resulting partition λ⁰⁰⁰ = (λo₁+λe₁, . . . , λo_r+λe_r).

Extension of Andrews’ identity

The remaining two cases are:

(2) If number of odd parts is greater than the number of even parts in a partition λ∈O_u^∗(n) and let the difference bet. Then a similar argument shows that thet rows in Y_λ⁰ remain left after adjoining of rows of Y_λ⁰ andY_λ⁰⁰. Therefore, in the resultingY_λ⁰⁰⁰ with

λ⁰⁰⁰ ∈P_eu^ou(n),t rows will be positioned in the same order as in Y_λ⁰. (3) Last, if the number of even parts is greater than the number of odd parts in a partition λ∈O_u^∗(n) and let the difference beu. Similarly, we see that urows inY_λ⁰⁰ remain left after adjoining the rows of Y_λ⁰ andY_λ⁰⁰ and hereu rows will be inserted intoY_λ⁰ so that the resulting Y_λ⁰⁰⁰ withλ⁰⁰⁰∈P_eu^ou(n) does not violate the structure of the Young diagram.

Koustav Banerjee BSP and Partition identities 38 / 45

Extension of Andrews’ identity

The remaining two cases are:

(2) If number of odd parts is greater than the number of even parts in a partition λ∈O_u^∗(n) and let the difference bet. Then a similar argument shows that thet rows in Y_λ⁰ remain left after adjoining of rows of Y_λ⁰ andY_λ⁰⁰. Therefore, in the resultingY_λ⁰⁰⁰ with

λ⁰⁰⁰ ∈P_eu^ou(n),t rows will be positioned in the same order as in Y_λ⁰.

(3) Last, if the number of even parts is greater than the number of odd parts in a partition λ∈O_u^∗(n) and let the difference beu. Similarly, we see that urows inY_λ⁰⁰ remain left after adjoining the rows of Y_λ⁰ andY_λ⁰⁰ and hereu rows will be inserted intoY_λ⁰ so that the resulting Y_λ⁰⁰⁰ withλ⁰⁰⁰∈P_eu^ou(n) does not violate the structure of the Young diagram.

Extension of Andrews’ identity

The remaining two cases are:

(2) If number of odd parts is greater than the number of even parts in a partition λ∈O_u^∗(n) and let the difference bet. Then a similar argument shows that thet rows in Y_λ⁰ remain left after adjoining of rows of Y_λ⁰ andY_λ⁰⁰. Therefore, in the resultingY_λ⁰⁰⁰ with

λ⁰⁰⁰ ∈P_eu^ou(n),t rows will be positioned in the same order as in Y_λ⁰. (3) Last, if the number of even parts is greater than the number of odd parts in a partition λ∈O_u^∗(n) and let the difference beu.

Similarly, we see that urows inY_λ⁰⁰ remain left after adjoining the rows of Y_λ⁰ andY_λ⁰⁰ and hereu rows will be inserted intoY_λ⁰ so that the resulting Y_λ⁰⁰⁰ withλ⁰⁰⁰ ∈P_eu^ou(n) does not violate the structure of the Young diagram.

Koustav Banerjee BSP and Partition identities 38 / 45

Extension of Andrews’ identity

For example, givenYλwith the partition λ= (5,4,3,2,1,1)∈O_u^∗(16):

Step 1: SeparatingYλ into the odd and even parts; i.e., into Y_λ⁰ with λ⁰ = (5,3,1,1) andY_λ⁰⁰ withλ⁰⁰ = (4,2) yields;

Extension of Andrews’ identity

For example, givenYλwith the partition λ= (5,4,3,2,1,1)∈O_u^∗(16):

Step 1: SeparatingYλ into the odd and even parts; i.e., into Y_λ⁰ with λ⁰ = (5,3,1,1) andY_λ⁰⁰ withλ⁰⁰ = (4,2) yields;

Koustav Banerjee BSP and Partition identities 39 / 45

Extension of Andrews’ identity

For example, givenYλwith the partition λ= (5,4,3,2,1,1)∈O_u^∗(16):

Step 1: SeparatingYλ into the odd and even parts; i.e., into Y_λ⁰ with λ⁰ = (5,3,1,1) andY_λ⁰⁰ withλ⁰⁰ = (4,2) yields;

Extension of Andrews’ identity

For example, givenYλwith the partition λ= (5,4,3,2,1,1)∈O_u^∗(16):

Step 1: SeparatingYλ into the odd and even parts; i.e., into Y_λ⁰ with λ⁰ = (5,3,1,1) andY_λ⁰⁰ withλ⁰⁰ = (4,2) yields;

Koustav Banerjee BSP and Partition identities 39 / 45

Extension of Andrews’ identity

Step 2: Adjoining the rows ofY_λ⁰ andY_λ⁰⁰ givesY_λ⁰⁰⁰ with the partition λ⁰⁰⁰ = (9,5,1,1)∈P_eu^ou(16);

Extension of Andrews’ identity

Step 2: Adjoining the rows ofY_λ⁰ andY_λ⁰⁰ givesY_λ⁰⁰⁰ with the partition λ⁰⁰⁰ = (9,5,1,1)∈P_eu^ou(16);

Koustav Banerjee BSP and Partition identities 40 / 45

Extension of Andrews’ identity

(1) All odd parts of µare distinct; i.e., there arei distinct odd values withµ_oi < µ_oi−1<· · ·< µ_o1. For allj (1≤j ≤i), we extract 2j−1 boxes from thejth row ofY_µ⁰ and attach 2j−1 boxes toY_µ⁰ without violating the structure of the Young diagram Y_µ⁰. Explicitly, we break an odd part µo_t of the partitionµ⁰ into

(µo_t−(2v−1),2v−1) where the part µo_t corresponds to the number of boxes in thevth row ofY_µ⁰. The Young diagramY_µ⁰⁰⁰ obtained from Y_µ⁰ by the above construction and adjoiningY_µ⁰⁰ with it to get the unique resulting Young diagram, sayYπ with π∈O_u^∗(n).

Extension of Andrews’ identity

(1) All odd parts of µare distinct; i.e., there arei distinct odd values withµ_oi < µ_oi−1<· · ·< µ_o1. For allj (1≤j ≤i), we extract 2j−1 boxes from thejth row ofY_µ⁰ and attach 2j−1 boxes toY_µ⁰ without violating the structure of the Young diagram Y_µ⁰. Explicitly, we break an odd part µo_t of the partitionµ⁰ into

(µo_t−(2v−1),2v−1) where the partµo_t corresponds to the number of boxes in thevth row ofY_µ⁰. The Young diagramY_µ⁰⁰⁰ obtained from Y_µ⁰ by the above construction and adjoiningY_µ⁰⁰ with it to get the unique resulting Young diagram, sayYπ with π∈O_u^∗(n).

Koustav Banerjee BSP and Partition identities 41 / 45

Extension of Andrews’ identity

(1) All odd parts of µare distinct; i.e., there arei distinct odd values withµ_oi < µ_oi−1<· · ·< µ_o1. For all j (1≤j ≤i), we extract 2j−1 boxes from thejth row ofY_µ⁰ and attach 2j−1 boxes toY_µ⁰ without violating the structure of the Young diagram Y_µ⁰. Explicitly, we break an odd part µo_t of the partitionµ⁰ into

(µo_t−(2v−1),2v−1) where the partµo_t corresponds to the number of boxes in thevth row ofY_µ⁰. The Young diagramY_µ⁰⁰⁰ obtained from Y_µ⁰ by the above construction and adjoiningY_µ⁰⁰ with it to get the unique resulting Young diagram, sayYπ with π∈O_u^∗(n).

Extension of Andrews’ identity

For example,Y_µ withµ= (9,7,4,2)∈P_eu^ou(22) breaks intoY_µ⁰ with µ⁰= (9,7) andY_µ⁰⁰ withµ⁰⁰= (4,2);

Step 1:

= +

Step 2: Following the above construction,Y_µ⁰ resultsY_µ⁰⁰⁰ with µ⁰⁰⁰= (1,3,6,6);

=

Step 3: Then the resulting diagramY_π with

π= (6,6,4,3,2,1)∈O_u^∗(22) is the unique pre-image of µ;

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Extension of Andrews’ identity

For example,Y_µ withµ= (9,7,4,2)∈P_eu^ou(22) breaks intoY_µ⁰ with µ⁰= (9,7) andY_µ⁰⁰ withµ⁰⁰= (4,2);

Step 1:

= +

Step 2: Following the above construction,Y_µ⁰ resultsY_µ⁰⁰⁰ with µ⁰⁰⁰= (1,3,6,6);

=

Step 3: Then the resulting diagramY_π with

π= (6,6,4,3,2,1)∈O_u^∗(22) is the unique pre-image of µ;

Extension of Andrews’ identity

For example,Y_µ withµ= (9,7,4,2)∈P_eu^ou(22) breaks intoY_µ⁰ with µ⁰= (9,7) andY_µ⁰⁰ withµ⁰⁰= (4,2);

Step 1:

= +

Step 2: Following the above construction,Y_µ⁰ resultsY_µ⁰⁰⁰ with µ⁰⁰⁰= (1,3,6,6);

=

Step 3: Then the resulting diagramY_π with

π= (6,6,4,3,2,1)∈O_u^∗(22) is the unique pre-image of µ;

Koustav Banerjee BSP and Partition identities 42 / 45

Extension of Andrews’ identity

For example,Y_µ withµ= (9,7,4,2)∈P_eu^ou(22) breaks intoY_µ⁰ with µ⁰= (9,7) andY_µ⁰⁰ withµ⁰⁰= (4,2);

Step 1:

= +

Step 2: Following the above construction,Y_µ⁰ resultsY_µ⁰⁰⁰ with µ⁰⁰⁰= (1,3,6,6);

=

Step 3: Then the resulting diagramY_π with

π= (6,6,4,3,2,1)∈O_u^∗(22) is the unique pre-image of µ;

Extension of Andrews’ identity

For example,Y_µ withµ= (9,7,4,2)∈P_eu^ou(22) breaks intoY_µ⁰ with µ⁰= (9,7) andY_µ⁰⁰ withµ⁰⁰= (4,2);

Step 1:

= +

Step 2: Following the above construction,Y_µ⁰ resultsY_µ⁰⁰⁰ with µ⁰⁰⁰= (1,3,6,6);

=

Step 3: Then the resulting diagramY_π with

π= (6,6,4,3,2,1)∈O_u^∗(22) is the unique pre-image of µ;

Koustav Banerjee BSP and Partition identities 42 / 45

Extension of Andrews’ identity

For example,Y_µ withµ= (9,7,4,2)∈P_eu^ou(22) breaks intoY_µ⁰ with µ⁰= (9,7) andY_µ⁰⁰ withµ⁰⁰= (4,2);

Step 1:

= +

Step 2: Following the above construction,Y_µ⁰ resultsY_µ⁰⁰⁰ with µ⁰⁰⁰= (1,3,6,6);

=

Step 3: Then the resulting diagramY_π with

π= (6,6,4,3,2,1)∈O_u^∗(22) is the unique pre-image of µ;

Extension of Andrews’ identity

Proof Sketch: (P_eu^ou(n)−→O_u^∗(n))

The remaining case:

(2) Odd parts of µrepeats; i.e.,µ⁰ = (µo₁, . . . , µo_i) with

µo_i < µoi−1 <· · ·< µo₁ with the assumption thatµo₁, . . . , µo_i occurs with multiplicityk1,k2, . . . ,ki, respectively. Now, for all 1≤t ≤i, we break thekt tuple (µo_t, . . . , µo_t) into

((µo_t−(2v−1),2v−1), . . . ,(µo_t−(2v−1),2v−1)), where the partµo_t corresponds to the number of boxes in thevth row ofY_µ⁰. Similar argument shows that the resulting partition, say π∈O_u^∗(n).

For example, the pre-image ofµ= (7,7,5,1,1,1)∈P_eu^ou(22) is π= (5,5,3,2,2,2,1,1,1)∈O_u^∗(22);

Koustav Banerjee BSP and Partition identities 43 / 45

Extension of Andrews’ identity

Proof Sketch: (P_eu^ou(n)−→O_u^∗(n)) The remaining case:

(2) Odd parts of µrepeats; i.e.,µ⁰ = (µo₁, . . . , µo_i) with

µo_i < µo_i−1 <· · ·< µo₁ with the assumption thatµo₁, . . . , µo_i occurs with multiplicityk1,k2, . . . ,ki, respectively. Now, for all 1≤t ≤i, we break thekt tuple (µo_t, . . . , µo_t) into

((µo_t−(2v−1),2v−1), . . . ,(µo_t−(2v−1),2v−1)), where the partµo_t corresponds to the number of boxes in thevth row ofY_µ⁰. Similar argument shows that the resulting partition, say π∈O_u^∗(n). For example, the pre-image ofµ= (7,7,5,1,1,1)∈P_eu^ou(22) is π= (5,5,3,2,2,2,1,1,1)∈O_u^∗(22);

Extension of Andrews’ identity

Proof Sketch: (P_eu^ou(n)−→O_u^∗(n)) The remaining case:

(2) Odd parts of µrepeats; i.e.,µ⁰ = (µo₁, . . . , µo_i) with

µo_i < µo_i−1 <· · ·< µo₁ with the assumption thatµo₁, . . . , µo_i occurs with multiplicityk1,k2, . . . ,ki, respectively. Now, for all 1≤t ≤i, we break thekt tuple (µo_t, . . . , µo_t) into

((µo_t−(2v−1),2v−1), . . . ,(µo_t−(2v−1),2v−1)), where the partµo_t corresponds to the number of boxes in thevth row ofY_µ⁰. Similar argument shows that the resulting partition, say π∈O_u^∗(n).

For example, the pre-image ofµ= (7,7,5,1,1,1)∈P_eu^ou(22) is π= (5,5,3,2,2,2,1,1,1)∈O_u^∗(22);

Koustav Banerjee BSP and Partition identities 43 / 45

Extension of Andrews’ identity

Proof Sketch: (P_eu^ou(n)−→O_u^∗(n)) The remaining case:

(2) Odd parts of µrepeats; i.e.,µ⁰ = (µo₁, . . . , µo_i) with

µo_i < µo_i−1 <· · ·< µo₁ with the assumption thatµo₁, . . . , µo_i occurs with multiplicityk1,k2, . . . ,ki, respectively. Now, for all 1≤t ≤i, we break thekt tuple (µo_t, . . . , µo_t) into

((µo_t−(2v−1),2v−1), . . . ,(µo_t−(2v−1),2v−1)), where the partµo_t corresponds to the number of boxes in thevth row ofY_µ⁰. Similar argument shows that the resulting partition, say π∈O_u^∗(n).

For example, the pre-image ofµ= (7,7,5,1,1,1)∈P_eu^ou(22) is π= (5,5,3,2,2,2,1,1,1)∈O_u^∗(22);

Extension of Andrews’ identity

Proof Sketch: (P_eu^ou(n)−→O_u^∗(n)) The remaining case:

(2) Odd parts of µrepeats; i.e.,µ⁰ = (µo₁, . . . , µo_i) with

µo_i < µo_i−1 <· · ·< µo₁ with the assumption thatµo₁, . . . , µo_i occurs with multiplicityk1,k2, . . . ,ki, respectively. Now, for all 1≤t ≤i, we break thekt tuple (µo_t, . . . , µo_t) into

((µo_t−(2v−1),2v−1), . . . ,(µo_t−(2v−1),2v−1)), where the partµo_t corresponds to the number of boxes in thevth row ofY_µ⁰. Similar argument shows that the resulting partition, say π∈O_u^∗(n).

For example, the pre-image ofµ= (7,7,5,1,1,1)∈P_eu^ou(22) is π= (5,5,3,2,2,2,1,1,1)∈O_u^∗(22);

Koustav Banerjee BSP and Partition identities 43 / 45

References

G.E. Andrews, The Theory of Partitions, Addison-Wesley Pub. Co., NY, 300 pp. (1976). Reissued, Cambridge University Press, New York, 1998.

C. Bessenrodt, On hooks of Young diagrams, Annals of Combinatorics2(1998), 103-110.

R. Bacher and L. Manivel,Hooks and powers of parts in partitions, S´eminaire Lotharingien de Combinatoire47(2002).

H.C. Chan,Ramanujan’s cubic continued fraction and an analogue of his most beautiful identity, International Journal of Number Theory06 (2010), 673–680.

M.G. Dastidar and S. Sengupta,Generalization of a few results in integer Partitions, Notes in Number theory and Discrete

Mathematics 19(2013), 69-76.

G.-N. Han, Some conjectures and open problems on partition hook lengths, Experimental Mathematics18(2009), 97-106.

R. Honsberger,Mathematical Gems III, Washington, DC: Math.

Thank you!

Koustav Banerjee BSP and Partition identities 45 / 45

In document Hook type tableaux and Partition identities (Stránka 111-156)