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 (λ^{0}) :λ^{0} ∈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 (λ^{0}) :λ^{0} ∈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 (λ^{0}) :λ^{0} ∈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 (λ^{0}) :λ^{0} ∈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 (λ^{0}) :λ^{0} ∈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 separateλintoλ^{0} = (λo_{1}, λo_{2}, . . . , λo_{r}) where 1≤oi≤`and
λ^{00}= (λe_{1}, λe_{2}, . . . , λe_{t}) where 1≤oj≤l according to the odd and
even parts, respectively with corresponding Young diagrams Y_{λ}^{0} and
Y_{λ}^{00}.

Next, we joinY_{λ}^{0} andY_{λ}^{00} by successively adjoining their rows with
respect to the ordering of the parts inλ^{0}, λ^{00}, respectively, starting
with the largest one and end with the smallest one with restricting
Young diagram, say,Y_{λ}^{000}.

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_{λ}^{000} is withλ^{000}∈P_{eu}^{ou}(n) as for
λ^{0} = (λo_{1}, . . . , λo_{r}) andλ^{00}= (λe_{1}, . . . , λe_{r}), the resulting partition
λ^{000} = (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 separateλintoλ^{0} = (λo_{1}, λo_{2}, . . . , λo_{r}) where 1≤oi≤`and
λ^{00}= (λe_{1}, λe_{2}, . . . , λe_{t}) where 1≤oj≤l according to the odd and
even parts, respectively with corresponding Young diagrams Y_{λ}^{0} and
Y_{λ}^{00}.

Next, we joinY_{λ}^{0} andY_{λ}^{00} by successively adjoining their rows with
respect to the ordering of the parts inλ^{0}, λ^{00}, respectively, starting
with the largest one and end with the smallest one with restricting
Young diagram, say,Y_{λ}^{000}.

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_{λ}^{000} is withλ^{000}∈P_{eu}^{ou}(n) as for
λ^{0} = (λo_{1}, . . . , λo_{r}) andλ^{00}= (λe_{1}, . . . , λe_{r}), the resulting partition
λ^{000} = (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 separateλintoλ^{0} = (λo_{1}, λo_{2}, . . . , λo_{r}) where 1≤oi≤`and
λ^{00}= (λe_{1}, λe_{2}, . . . , λe_{t}) where 1≤oj≤l according to the odd and
even parts, respectively with corresponding Young diagrams Y_{λ}^{0} and
Y_{λ}^{00}.

Next, we joinY_{λ}^{0} andY_{λ}^{00} by successively adjoining their rows with
respect to the ordering of the parts inλ^{0}, λ^{00}, respectively, starting
with the largest one and end with the smallest one with restricting
Young diagram, say,Y_{λ}^{000}.

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_{λ}^{000} is withλ^{000}∈P_{eu}^{ou}(n) as for
λ^{0} = (λo_{1}, . . . , λo_{r}) andλ^{00}= (λe_{1}, . . . , λe_{r}), the resulting partition
λ^{000} = (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).

^{0} = (λo_{1}, λo_{2}, . . . , λo_{r}) where 1≤oi≤`and
λ^{00}= (λe_{1}, λe_{2}, . . . , λe_{t}) where 1≤oj≤l according to the odd and
even parts, respectively with corresponding Young diagrams Y_{λ}^{0} and
Y_{λ}^{00}.

_{λ}^{0} andY_{λ}^{00} by successively adjoining their rows with
respect to the ordering of the parts inλ^{0}, λ^{00}, respectively, starting
with the largest one and end with the smallest one with restricting
Young diagram, say,Y_{λ}^{000}.

Now, we consider the following three cases:

_{u}^{∗}(n), then Y_{λ}^{000} is withλ^{000}∈P_{eu}^{ou}(n) as for
λ^{0} = (λo_{1}, . . . , λo_{r}) andλ^{00}= (λe_{1}, . . . , λe_{r}), the resulting partition
λ^{000} = (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).

^{0} = (λo_{1}, λo_{2}, . . . , λo_{r}) where 1≤oi≤`and
λ^{00}= (λe_{1}, λe_{2}, . . . , λe_{t}) where 1≤oj≤l according to the odd and
even parts, respectively with corresponding Young diagrams Y_{λ}^{0} and
Y_{λ}^{00}.

_{λ}^{0} andY_{λ}^{00} by successively adjoining their rows with
respect to the ordering of the parts inλ^{0}, λ^{00}, respectively, starting
with the largest one and end with the smallest one with restricting
Young diagram, say,Y_{λ}^{000}.

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_{λ}^{000} is withλ^{000}∈P_{eu}^{ou}(n) as for
λ^{0} = (λo_{1}, . . . , λo_{r}) andλ^{00}= (λe_{1}, . . . , λe_{r}), the resulting partition
λ^{000} = (λo_{1}+λe_{1}, . . . , 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_{λ}^{0} remain left after adjoining of
rows of Y_{λ}^{0} andY_{λ}^{00}. Therefore, in the resultingY_{λ}^{000} with

λ^{000} ∈P_{eu}^{ou}(n),t rows will be positioned in the same order as in Y_{λ}^{0}.
(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_{λ}^{00} remain left after adjoining the
rows of Y_{λ}^{0} andY_{λ}^{00} and hereu rows will be inserted intoY_{λ}^{0} so
that the resulting Y_{λ}^{000} withλ^{000}∈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_{λ}^{0} remain left after adjoining of
rows of Y_{λ}^{0} andY_{λ}^{00}. Therefore, in the resultingY_{λ}^{000} with

λ^{000} ∈P_{eu}^{ou}(n),t rows will be positioned in the same order as in Y_{λ}^{0}.

(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_{λ}^{00} remain left after adjoining the
rows of Y_{λ}^{0} andY_{λ}^{00} and hereu rows will be inserted intoY_{λ}^{0} so
that the resulting Y_{λ}^{000} withλ^{000}∈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_{λ}^{0} remain left after adjoining of
rows of Y_{λ}^{0} andY_{λ}^{00}. Therefore, in the resultingY_{λ}^{000} with

λ^{000} ∈P_{eu}^{ou}(n),t rows will be positioned in the same order as in Y_{λ}^{0}.
(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_{λ}^{00} remain left after adjoining the
rows of Y_{λ}^{0} andY_{λ}^{00} and hereu rows will be inserted intoY_{λ}^{0} so
that the resulting Y_{λ}^{000} withλ^{000} ∈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_{λ}^{0} with
λ^{0} = (5,3,1,1) andY_{λ}^{00} withλ^{00} = (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_{λ}^{0} with
λ^{0} = (5,3,1,1) andY_{λ}^{00} withλ^{00} = (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_{λ}^{0} with
λ^{0} = (5,3,1,1) andY_{λ}^{00} withλ^{00} = (4,2) yields;

### Extension of Andrews’ identity

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

_{λ}^{0} with
λ^{0} = (5,3,1,1) andY_{λ}^{00} withλ^{00} = (4,2) yields;

Koustav Banerjee BSP and Partition identities 39 / 45

### Extension of Andrews’ identity

Step 2: Adjoining the rows ofY_{λ}^{0} andY_{λ}^{00} givesY_{λ}^{000} with the partition
λ^{000} = (9,5,1,1)∈P_{eu}^{ou}(16);

### Extension of Andrews’ identity

Step 2: Adjoining the rows ofY_{λ}^{0} andY_{λ}^{00} givesY_{λ}^{000} with the partition
λ^{000} = (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_{o}_{i} < µ_{o}_{i−1}<· · ·< µ_{o}_{1}. For allj (1≤j ≤i), we extract
2j−1 boxes from thejth row ofY_{µ}^{0} and attach 2j−1 boxes toY_{µ}^{0}
without violating the structure of the Young diagram Y_{µ}^{0}. Explicitly,
we break an odd part µo_{t} of the partitionµ^{0} into

(µo_{t}−(2v−1),2v−1) where the part µo_{t} corresponds to the
number of boxes in thevth row ofY_{µ}^{0}. The Young diagramY_{µ}^{000}
obtained from Y_{µ}^{0} by the above construction and adjoiningY_{µ}^{00}
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_{o}_{i} < µ_{o}_{i−1}<· · ·< µ_{o}_{1}. For allj (1≤j ≤i), we extract
2j−1 boxes from thejth row ofY_{µ}^{0} and attach 2j−1 boxes toY_{µ}^{0}
without violating the structure of the Young diagram Y_{µ}^{0}. Explicitly,
we break an odd part µo_{t} of the partitionµ^{0} into

(µo_{t}−(2v−1),2v−1) where the partµo_{t} corresponds to the
number of boxes in thevth row ofY_{µ}^{0}. The Young diagramY_{µ}^{000}
obtained from Y_{µ}^{0} by the above construction and adjoiningY_{µ}^{00}
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_{o}_{i} < µ_{o}_{i−1}<· · ·< µ_{o}_{1}. For all j (1≤j ≤i), we extract
2j−1 boxes from thejth row ofY_{µ}^{0} and attach 2j−1 boxes toY_{µ}^{0}
without violating the structure of the Young diagram Y_{µ}^{0}. Explicitly,
we break an odd part µo_{t} of the partitionµ^{0} into

(µo_{t}−(2v−1),2v−1) where the partµo_{t} corresponds to the
number of boxes in thevth row ofY_{µ}^{0}. The Young diagramY_{µ}^{000}
obtained from Y_{µ}^{0} by the above construction and adjoiningY_{µ}^{00}
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_{µ}^{0} with
µ^{0}= (9,7) andY_{µ}^{00} withµ^{00}= (4,2);

Step 1:

= +

Step 2: Following the above construction,Y_{µ}^{0} resultsY_{µ}^{000} with
µ^{000}= (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_{µ}^{0} with
µ^{0}= (9,7) andY_{µ}^{00} withµ^{00}= (4,2);

Step 1:

= +

Step 2: Following the above construction,Y_{µ}^{0} resultsY_{µ}^{000} with
µ^{000}= (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_{µ}^{0} with
µ^{0}= (9,7) andY_{µ}^{00} withµ^{00}= (4,2);

Step 1:

= +

Step 2: Following the above construction,Y_{µ}^{0} resultsY_{µ}^{000} with
µ^{000}= (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

_{µ} withµ= (9,7,4,2)∈P_{eu}^{ou}(22) breaks intoY_{µ}^{0} with
µ^{0}= (9,7) andY_{µ}^{00} withµ^{00}= (4,2);

Step 1:

= +

Step 2: Following the above construction,Y_{µ}^{0} resultsY_{µ}^{000} with
µ^{000}= (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

_{µ} withµ= (9,7,4,2)∈P_{eu}^{ou}(22) breaks intoY_{µ}^{0} with
µ^{0}= (9,7) andY_{µ}^{00} withµ^{00}= (4,2);

Step 1:

= +

Step 2: Following the above construction,Y_{µ}^{0} resultsY_{µ}^{000} with
µ^{000}= (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

_{µ} withµ= (9,7,4,2)∈P_{eu}^{ou}(22) breaks intoY_{µ}^{0} with
µ^{0}= (9,7) andY_{µ}^{00} withµ^{00}= (4,2);

Step 1:

= +

Step 2: Following the above construction,Y_{µ}^{0} resultsY_{µ}^{000} with
µ^{000}= (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.,µ^{0} = (µo_{1}, . . . , µo_{i}) with

µo_{i} < µoi−1 <· · ·< µo_{1} with the assumption thatµo_{1}, . . . , 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_{µ}^{0}.
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.,µ^{0} = (µo_{1}, . . . , µo_{i}) with

µo_{i} < µo_{i−1} <· · ·< µo_{1} with the assumption thatµo_{1}, . . . , 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_{µ}^{0}.
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.,µ^{0} = (µo_{1}, . . . , µo_{i}) with

µo_{i} < µo_{i−1} <· · ·< µo_{1} with the assumption thatµo_{1}, . . . , 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_{µ}^{0}.
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.,µ^{0} = (µo_{1}, . . . , µo_{i}) with

µo_{i} < µo_{i−1} <· · ·< µo_{1} with the assumption thatµo_{1}, . . . , 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_{µ}^{0}.
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.,µ^{0} = (µo_{1}, . . . , µo_{i}) with

_{i} < µo_{i−1} <· · ·< µo_{1} with the assumption thatµo_{1}, . . . , µo_{i} occurs
with multiplicityk1,k2, . . . ,ki, respectively. Now, for all 1≤t ≤i,
we break thekt tuple (µo_{t}, . . . , µo_{t}) into

_{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_{µ}^{0}.
Similar argument shows that the resulting partition, say π∈O_{u}^{∗}(n).

_{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