Theorem (Stanley (1972))

The total number of1’s in all partitions of a positive integer n is equal to the sum of the numbers of distinct parts of all partitions of n; i.e., S(n) =Q1(n).

Koustav Banerjee BSP and Partition identities 8 / 45

### Two partition identities

Theorem (Stanley (1972))

The total number of1’s in all partitions of a positive integer n is equal to the sum of the numbers of distinct parts of all partitions of n; i.e., S(n) =Q1(n).

Forn= 4,

P(4)

λ`4 δ1(λ) µ(λ) B_{(0,0)}(λ)

4 0 1 1

3 + 1 1 2 2

2 + 2 0 1 1

2 + 1 + 1 2 2 2

1 + 1 + 1 + 1 4 1 1

Total Q1(4) = 7 S(4) = 7 B_{(0,0)}(n) = 7
.

### Two partition identities

Proof sketch:

We will show that number of distinct parts of a partition λ`nis
equal to the number of boxes in Y_{λ} with hook-type (0,0); i.e.,
µ(λ) =B_{(0,0)}(λ).

Let λ= (λ1, λ2, . . . , λr)`nand supposeλa_{1}, λa_{2}, . . . , λa_{k} are all the
distinct parts of λwith respective multiplicitiesm1,m2, . . . ,mk

where 1≤ai ≤r andai ∈Nfor all 1≤i≤k. Without loss of
generality assumeλa1 > λa2>· · ·> λa_{k}.

Next, we note that, the boxes with hook-type (0,0) appear exactly
once inYλ corresponding to the partλa_{m} subject to the condition
that the immediate next part λa_{n} withm6=n.

Therefore, the number of boxes with hook-type (0,0) equals the number of distinct parts ofλ. Now, summing over allλ`nwe get the Stanley’s theorem.

Koustav Banerjee BSP and Partition identities 9 / 45

### Two partition identities

Proof sketch:

We will show that number of distinct parts of a partition λ`nis
equal to the number of boxes in Y_{λ} with hook-type (0,0); i.e.,
µ(λ) =B_{(0,0)}(λ).

Let λ= (λ1, λ2, . . . , λr)`nand supposeλa_{1}, λa_{2}, . . . , λa_{k} are all the
distinct parts of λwith respective multiplicitiesm1,m2, . . . ,mk

where 1≤ai ≤r andai ∈Nfor all 1≤i≤k. Without loss of
generality assumeλa1 > λa2>· · ·> λa_{k}.

Next, we note that, the boxes with hook-type (0,0) appear exactly
once inYλ corresponding to the partλa_{m} subject to the condition
that the immediate next part λa_{n} withm6=n.

Therefore, the number of boxes with hook-type (0,0) equals the number of distinct parts ofλ. Now, summing over allλ`nwe get the Stanley’s theorem.

### Two partition identities

Proof sketch:

We will show that number of distinct parts of a partition λ`nis
equal to the number of boxes in Y_{λ} with hook-type (0,0); i.e.,
µ(λ) =B_{(0,0)}(λ).

Let λ= (λ1, λ2, . . . , λr)`nand supposeλa_{1}, λa_{2}, . . . , λa_{k} are all the
distinct parts of λwith respective multiplicitiesm1,m2, . . . ,mk

where 1≤ai ≤r andai ∈Nfor all 1≤i≤k. Without loss of
generality assumeλa1 > λa2>· · ·> λa_{k}.

Next, we note that, the boxes with hook-type (0,0) appear exactly
once inYλ corresponding to the partλa_{m} subject to the condition
that the immediate next part λa_{n} withm6=n.

Therefore, the number of boxes with hook-type (0,0) equals the number of distinct parts ofλ. Now, summing over allλ`nwe get the Stanley’s theorem.

Koustav Banerjee BSP and Partition identities 9 / 45

### Two partition identities

Proof sketch:

_{λ} with hook-type (0,0); i.e.,
µ(λ) =B_{(0,0)}(λ).

_{1}, λa_{2}, . . . , λa_{k} are all the
distinct parts of λwith respective multiplicitiesm1,m2, . . . ,mk

where 1≤ai ≤r andai ∈Nfor all 1≤i≤k. Without loss of
generality assumeλa1 > λa2>· · ·> λa_{k}.

_{m} subject to the condition
that the immediate next part λa_{n} withm6=n.

### Two partition identities

Proof sketch:

_{λ} with hook-type (0,0); i.e.,
µ(λ) =B_{(0,0)}(λ).

_{1}, λa_{2}, . . . , λa_{k} are all the
distinct parts of λwith respective multiplicitiesm1,m2, . . . ,mk

where 1≤ai ≤r andai ∈Nfor all 1≤i≤k. Without loss of
generality assumeλa1 > λa2>· · ·> λa_{k}.

_{m} subject to the condition
that the immediate next part λa_{n} withm6=n.

Koustav Banerjee BSP and Partition identities 9 / 45

### Two partition identities

Theorem (Elder (1984))

The total number of occurences of an integer k among all partitions of n is equal to the number of occasions that a part occurs greater or equal k times in P(n); i.e., Qk(n) =Vk(n).

### Two partition identities

Theorem (Elder (1984))

The total number of occurences of an integer k among all partitions of n is equal to the number of occasions that a part occurs greater or equal k times in P(n); i.e., Qk(n) =Vk(n).

Koustav Banerjee BSP and Partition identities 10 / 45

### Two partition identities

Proof sketch:

We need only to show that the number of boxes with hook type (k−1,0), k>1, in a partitionλ`nis equal to the number of parts that occur k or more times in λ.

Now, a box with hook-type (k−1,0) inYλ with

λ= (λ_{1}, λ_{2}, . . . , λ_{r})`nprecisely describes that there arek−1
boxes on the right to it but having no box above.

When transformingλto it’s conjugateλ^{0} it is clear that after
conjugation, the box with hook-type (k−1,0) transforms into the
box with hook-type (0,k−1). This shows that there are total at
leastk verticals stacks of boxes (including the box itself); i.e., there
exists a part that occurs at leastk times in that conjugate partition.
So, corresponding to each box with hook-type (k−1,0) there exists
a part that occurs at leastk times and hence summing over all
partitions of nwe have Elder’s statement.

### Two partition identities

Proof sketch:

We need only to show that the number of boxes with hook type (k−1,0), k>1, in a partitionλ`nis equal to the number of parts that occur k or more times in λ.

Now, a box with hook-type (k−1,0) inYλ with

λ= (λ_{1}, λ_{2}, . . . , λ_{r})`nprecisely describes that there arek−1
boxes on the right to it but having no box above.

When transformingλto it’s conjugateλ^{0} it is clear that after
conjugation, the box with hook-type (k−1,0) transforms into the
box with hook-type (0,k−1). This shows that there are total at
leastk verticals stacks of boxes (including the box itself); i.e., there
exists a part that occurs at leastk times in that conjugate partition.
So, corresponding to each box with hook-type (k−1,0) there exists
a part that occurs at leastk times and hence summing over all
partitions of nwe have Elder’s statement.

Koustav Banerjee BSP and Partition identities 11 / 45

### Two partition identities

Proof sketch:

We need only to show that the number of boxes with hook type (k−1,0), k>1, in a partitionλ`nis equal to the number of parts that occur k or more times in λ.

Now, a box with hook-type (k−1,0) inYλ with

λ= (λ1, λ2, . . . , λr)`nprecisely describes that there arek−1 boxes on the right to it but having no box above.

When transformingλto it’s conjugateλ^{0} it is clear that after
conjugation, the box with hook-type (k−1,0) transforms into the
box with hook-type (0,k−1). This shows that there are total at
leastk verticals stacks of boxes (including the box itself); i.e., there
exists a part that occurs at leastk times in that conjugate partition.
So, corresponding to each box with hook-type (k−1,0) there exists
a part that occurs at leastk times and hence summing over all
partitions of nwe have Elder’s statement.

### Two partition identities

Proof sketch:

Now, a box with hook-type (k−1,0) inYλ with

λ= (λ1, λ2, . . . , λr)`nprecisely describes that there arek−1 boxes on the right to it but having no box above.

When transformingλto it’s conjugateλ^{0} it is clear that after
conjugation, the box with hook-type (k−1,0) transforms into the
box with hook-type (0,k−1). This shows that there are total at
leastk verticals stacks of boxes (including the box itself); i.e., there
exists a part that occurs at leastk times in that conjugate partition.

So, corresponding to each box with hook-type (k−1,0) there exists a part that occurs at leastk times and hence summing over all partitions of nwe have Elder’s statement.

Koustav Banerjee BSP and Partition identities 11 / 45

### Two partition identities

Proof sketch:

Now, a box with hook-type (k−1,0) inYλ with

λ= (λ1, λ2, . . . , λr)`nprecisely describes that there arek−1 boxes on the right to it but having no box above.

When transformingλto it’s conjugateλ^{0} it is clear that after
conjugation, the box with hook-type (k−1,0) transforms into the
box with hook-type (0,k−1). This shows that there are total at
leastk verticals stacks of boxes (including the box itself); i.e., there
exists a part that occurs at leastk times in that conjugate partition.

So, corresponding to each box with hook-type (k−1,0) there exists a part that occurs at leastk times and hence summing over all partitions of nwe have Elder’s statement.

### Box Stacking Principle (BSP)

The BSP consists of a set of rules to produce from all partitions ofna new set of partitions ofn+k wherek is a positive integer. Given a partitionλ`n, the new partitions are produced by addingk boxes as follows:

Fork = 1:

We add one box to all permissible places inYλ. One can trivially add
one box in two ways: (i) Add to the bottom row ofYλ. (ii) Stack
the box on the above of the top row ofYλ. Also, we can add one
box to a row inYλ if and only if the difference between the number
of boxes in the chosen row and its immediate next is at least 1.
Explicitly, for λ:= (λ_{1}, . . . , λ_{r})`n, following rule (i) the trivial
addition of one box corresponds to µ:= ((λ1+ 1), . . . , λr)`n+ 1
whereas by rule (ii) we haveµ:= (λ1, . . . , λr,1)`n+ 1. Nontrivial
addition of one box can be done if and only if for any two

consecutive part say, λi andλj (λi ≥λj), we haveλi−λj ≥1.

Koustav Banerjee BSP and Partition identities 12 / 45

### Box Stacking Principle (BSP)

The BSP consists of a set of rules to produce from all partitions ofna new set of partitions ofn+k wherek is a positive integer. Given a partitionλ`n, the new partitions are produced by addingk boxes as follows:

Fork = 1:

We add one box to all permissible places inYλ. One can trivially add
one box in two ways: (i) Add to the bottom row ofYλ. (ii) Stack
the box on the above of the top row ofYλ. Also, we can add one
box to a row inYλ if and only if the difference between the number
of boxes in the chosen row and its immediate next is at least 1.
Explicitly, for λ:= (λ_{1}, . . . , λ_{r})`n, following rule (i) the trivial
addition of one box corresponds to µ:= ((λ1+ 1), . . . , λr)`n+ 1
whereas by rule (ii) we haveµ:= (λ1, . . . , λr,1)`n+ 1. Nontrivial
addition of one box can be done if and only if for any two

consecutive part say, λi andλj (λi ≥λj), we haveλi−λj ≥1.

### Box Stacking Principle (BSP)

The BSP consists of a set of rules to produce from all partitions ofna new set of partitions ofn+k wherek is a positive integer. Given a partitionλ`n, the new partitions are produced by addingk boxes as follows:

Fork = 1:

We add one box to all permissible places inYλ. One can trivially add one box in two ways: (i) Add to the bottom row ofYλ. (ii) Stack the box on the above of the top row ofYλ. Also, we can add one box to a row inYλ if and only if the difference between the number of boxes in the chosen row and its immediate next is at least 1.

Explicitly, for λ:= (λ_{1}, . . . , λ_{r})`n, following rule (i) the trivial
addition of one box corresponds to µ:= ((λ1+ 1), . . . , λr)`n+ 1
whereas by rule (ii) we haveµ:= (λ1, . . . , λr,1)`n+ 1. Nontrivial
addition of one box can be done if and only if for any two

consecutive part say, λi andλj (λi ≥λj), we haveλi−λj ≥1.

Koustav Banerjee BSP and Partition identities 12 / 45

### Box Stacking Principle (BSP)

Fork = 1:

We add one box to all permissible places inYλ. One can trivially add one box in two ways: (i) Add to the bottom row ofYλ. (ii) Stack the box on the above of the top row ofYλ. Also, we can add one box to a row inYλ if and only if the difference between the number of boxes in the chosen row and its immediate next is at least 1.

Explicitly, for λ:= (λ_{1}, . . . , λ_{r})`n, following rule (i) the trivial
addition of one box corresponds to µ:= ((λ_{1}+ 1), . . . , λ_{r})`n+ 1
whereas by rule (ii) we haveµ:= (λ1, . . . , λr,1)`n+ 1. Nontrivial
addition of one box can be done if and only if for any two

consecutive part say, λi andλj (λi ≥λj), we haveλi−λj ≥1.

### Box Stacking Principle (BSP)

For example, to all partitions of 4 and applying the stacking principle for adding one box to the Young diagram gives:

I.λ= 4 :

+ =

= II.λ= 3 + 1 :

+ =

=

=

Koustav Banerjee BSP and Partition identities 13 / 45

### Box Stacking Principle (BSP)

For example, to all partitions of 4 and applying the stacking principle for adding one box to the Young diagram gives:

I.λ= 4 :

+ =

=

II.λ= 3 + 1 :

+ =

=

=

### Box Stacking Principle (BSP)

For example, to all partitions of 4 and applying the stacking principle for adding one box to the Young diagram gives:

I.λ= 4 :

+ =

= II.λ= 3 + 1 :

+ =

=

=

Koustav Banerjee BSP and Partition identities 13 / 45

### Box Stacking Principle (BSP)

III.λ= 2 + 2 :

+ =

=

IV.λ= 2 + 1 + 1 :

+ =

=

### Box Stacking Principle (BSP)

III.λ= 2 + 2 :

+ =

=

IV.λ= 2 + 1 + 1 :

+ =

=

Koustav Banerjee BSP and Partition identities 14 / 45

### Box Stacking Principle (BSP)

IV.λ= 2 + 1 + 1 :

+ =

V.λ= 1 + 1 + 1 + 1 :

+ =

=

### Box Stacking Principle (BSP)

IV.λ= 2 + 1 + 1 :

+ =

V.λ= 1 + 1 + 1 + 1 :

+ =

=

Koustav Banerjee BSP and Partition identities 15 / 45

### Box Stacking Principle (BSP)

Fork >1:

Here we consider the addition ofk boxes as a ‘packet ofk boxes’, instead of adding ‘k single boxes’. Again one can trivially add a

‘packet ofk boxes’ to the bottom row ofY_{λ} withλ:= (λ_{1}, . . . , λ_{r}).
By adding ‘packet ofk boxes’, we mean that addingk toλ_{1}so that
the resulting partition µ:= ((λ_{1}+k), . . . , λ_{r})`n+k.

A nontrivial addition of a packet of k boxes toYλ can be done if
and only if for any two consecutive part say, λi andλj (λi ≥λj), we
have λ_{i}−λ_{j} ≥k.

For stacking ofk = 2 boxes withλ= (3,1)`4 following BSP,

+ =

=

### Box Stacking Principle (BSP)

Fork >1:

Here we consider the addition ofk boxes as a ‘packet ofk boxes’, instead of adding ‘k single boxes’. Again one can trivially add a

‘packet ofk boxes’ to the bottom row ofY_{λ} withλ:= (λ_{1}, . . . , λ_{r}).

By adding ‘packet ofk boxes’, we mean that addingk toλ_{1} so that
the resulting partition µ:= ((λ_{1}+k), . . . , λ_{r})`n+k.

A nontrivial addition of a packet of k boxes toYλ can be done if
and only if for any two consecutive part say, λi andλj (λi ≥λj), we
have λ_{i}−λ_{j} ≥k.

For stacking ofk = 2 boxes withλ= (3,1)`4 following BSP,

+ =

=

Koustav Banerjee BSP and Partition identities 16 / 45

### Box Stacking Principle (BSP)

Fork >1:

Here we consider the addition ofk boxes as a ‘packet ofk boxes’, instead of adding ‘k single boxes’. Again one can trivially add a

‘packet ofk boxes’ to the bottom row ofY_{λ} withλ:= (λ_{1}, . . . , λ_{r}).

By adding ‘packet ofk boxes’, we mean that addingk toλ_{1} so that
the resulting partition µ:= ((λ_{1}+k), . . . , λ_{r})`n+k.

A nontrivial addition of a packet of k boxes toYλ can be done if
and only if for any two consecutive part say, λi andλj (λi ≥λj), we
have λ_{i}−λ_{j} ≥k.

For stacking ofk = 2 boxes withλ= (3,1)`4 following BSP,

+ =

=

### Box Stacking Principle (BSP)

Fork >1:

‘packet ofk boxes’ to the bottom row ofY_{λ} withλ:= (λ_{1}, . . . , λ_{r}).

By adding ‘packet ofk boxes’, we mean that addingk toλ_{1} so that
the resulting partition µ:= ((λ_{1}+k), . . . , λ_{r})`n+k.

_{i}−λ_{j} ≥k.

For stacking ofk= 2 boxes withλ= (3,1)`4 following BSP,

+ =

=

Koustav Banerjee BSP and Partition identities 16 / 45

### Box Stacking Principle (BSP)

What we do not allow:

We do not consider the addition of ‘k single boxes’ which means that
we do not allow the cases µ1:= (λ1, . . . , λr,1, . . . ,1)`n+k and
µ2:= (λ1, . . . ,(λj_{1}+ 1), . . . ,(λj_{2}+ 1), . . . , ,(λj_{k}+ 1), . . . , λr)`n+k.
For stacking ofk = 2 boxes withλ= (3,1)`4, following situations will
be regarded as violating our rules;

+ =

=

=

### Box Stacking Principle (BSP)

What we do not allow:

We do not consider the addition of ‘k single boxes’ which means that
we do not allow the cases µ1:= (λ1, . . . , λr,1, . . . ,1)`n+k and
µ2:= (λ1, . . . ,(λj_{1}+ 1), . . . ,(λj_{2}+ 1), . . . , ,(λj_{k}+ 1), . . . , λr)`n+k.

For stacking ofk = 2 boxes withλ= (3,1)`4, following situations will be regarded as violating our rules;

+ =

=

=

Koustav Banerjee BSP and Partition identities 17 / 45

### Box Stacking Principle (BSP)

What we do not allow:

We do not consider the addition of ‘k single boxes’ which means that
we do not allow the cases µ1:= (λ1, . . . , λr,1, . . . ,1)`n+k and
µ2:= (λ1, . . . ,(λj_{1}+ 1), . . . ,(λj_{2}+ 1), . . . , ,(λj_{k}+ 1), . . . , λr)`n+k.
For stacking ofk= 2 boxes withλ= (3,1)`4, following situations will
be regarded as violating our rules;

+ =

=

=

### Box Stacking Principle (BSP)

What we do not allow:

We do not consider the addition of ‘k single boxes’ which means that
we do not allow the cases µ1:= (λ1, . . . , λr,1, . . . ,1)`n+k and
µ2:= (λ1, . . . ,(λj_{1}+ 1), . . . ,(λj_{2}+ 1), . . . , ,(λj_{k}+ 1), . . . , λr)`n+k.
For stacking ofk= 2 boxes withλ= (3,1)`4, following situations will
be regarded as violating our rules;

+ =

=

=

Koustav Banerjee BSP and Partition identities 17 / 45