The BSP in color partitions context consists of a set of rules to produce
from all color partitionsλofna new set of color partitions ofn+k. In
this context we have to take care about the color of a ‘packet ofk
boxes’. Ifk is not a multiple of`, without loss of generality, we always
add a ‘packet ofk boxes’ prescribed by white color. The set of rules are
as follows:
bottom row of Y_{λ}so that the resulting partition

µ:= ((λ_{1}_{i}

1 +k_{1}), . . . , λ_{r}_{ir})∈P^{(`)}(n+k).

Koustav Banerjee BSP and Partition identities 20 / 45

### BSP in color partitions

The BSP in color partitions context consists of a set of rules to produce
from all color partitionsλofna new set of color partitions ofn+k. In
this context we have to take care about the color of a ‘packet ofk
boxes’. Ifk is not a multiple of`, without loss of generality, we always
add a ‘packet ofk boxes’ prescribed by white color. The set of rules are
as follows:
bottom row of Y_{λ}so that the resulting partition

µ:= ((λ_{1}_{i}

1 +k_{1}), . . . , λ_{r}_{ir})∈P^{(`)}(n+k).

### BSP in color partitions

The BSP in color partitions context consists of a set of rules to produce
from all color partitionsλofna new set of color partitions ofn+k. In
this context we have to take care about the color of a ‘packet ofk
boxes’. Ifk is not a multiple of`, without loss of generality, we always
add a ‘packet ofk boxes’ prescribed by white color. The set of rules are
as follows:
bottom row of Y_{λ} so that the resulting partition

µ:= ((λ_{1}_{i}

1 +k_{1}), . . . , λ_{r}_{ir})∈P^{(`)}(n+k).

Koustav Banerjee BSP and Partition identities 20 / 45

### BSP in color partitions

Ifi1= 2, then two cases will arise:

A.Ifλ_{1}≥k, then we consider following two cases:

(i) If there exist any two consecutive parts sayλ_{s}_{is} andλ_{t}_{it}
(λ_{s} ≥λ_{t}) withi_{t} = 1 andλ_{s}−λ_{t} ≥k, then we add a packet of k boxes
to the row corresponding to the partλt_{it} in Yλ.

(ii) If there does not exists any two consecutive parts with the condition given in (i), then we simply insert the packet ofk-boxes as a new row intoYλ.

For example, if we consider the addition of a packet of 3 boxes to the
partitionλ= (32,11)∈P^{(3)}(4), then:

+ =

### BSP in color partitions

Ifi1= 2, then two cases will arise:

A.Ifλ_{1}≥k, then we consider following two cases:

(i) If there exist any two consecutive parts sayλ_{s}_{is} andλ_{t}_{it}
(λ_{s} ≥λ_{t}) withi_{t} = 1 andλ_{s} −λ_{t} ≥k, then we add a packet of k boxes
to the row corresponding to the partλt_{it} in Yλ.

(ii) If there does not exists any two consecutive parts with the condition given in (i), then we simply insert the packet ofk-boxes as a new row intoYλ.

For example, if we consider the addition of a packet of 3 boxes to the
partitionλ= (32,11)∈P^{(3)}(4), then:

+ =

Koustav Banerjee BSP and Partition identities 21 / 45

### BSP in color partitions

Ifi1= 2, then two cases will arise:

A.Ifλ_{1}≥k, then we consider following two cases:

(i) If there exist any two consecutive parts sayλ_{s}_{is} andλ_{t}_{it}
(λ_{s} ≥λ_{t}) withi_{t} = 1 andλ_{s} −λ_{t} ≥k, then we add a packet ofk boxes
to the row corresponding to the partλ_{t}_{it} in Y_{λ}.

(ii) If there does not exists any two consecutive parts with the condition given in (i), then we simply insert the packet ofk-boxes as a new row intoYλ.

For example, if we consider the addition of a packet of 3 boxes to the
partitionλ= (32,11)∈P^{(3)}(4), then:

+ =

### BSP in color partitions

Ifi1= 2, then two cases will arise:

A.Ifλ_{1}≥k, then we consider following two cases:

(i) If there exist any two consecutive parts sayλ_{s}_{is} andλ_{t}_{it}
(λ_{s} ≥λ_{t}) withi_{t} = 1 andλ_{s} −λ_{t} ≥k, then we add a packet ofk boxes
to the row corresponding to the partλ_{t}_{it} in Y_{λ}.

^{(3)}(4), then:

+ =

Koustav Banerjee BSP and Partition identities 21 / 45

### BSP in color partitions

Ifi1= 2, then two cases will arise:

A.Ifλ_{1}≥k, then we consider following two cases:

(i) If there exist any two consecutive parts sayλ_{s}_{is} andλ_{t}_{it}
(λ_{s} ≥λ_{t}) withi_{t} = 1 andλ_{s} −λ_{t} ≥k, then we add a packet ofk boxes
to the row corresponding to the partλ_{t}_{it} in Y_{λ}.

^{(3)}(4), then:

+ =

### BSP in color partitions

B.Ifλ_{1}<k, then we adjoin the packet ofk boxes to the below of
the bottom row ofY_{λ} so that resulting partition is

µ:= (k_{1}, λ_{1}_{i}

1, . . . , λ_{r}_{ir})∈P^{(`)}(n+k).

For example, if we consider the addition of a packet of 5 boxes to the
partitionλ= (32,11)∈P^{(3)}(4), then:

+ =

Koustav Banerjee BSP and Partition identities 22 / 45

### BSP in color partitions

B.Ifλ_{1}<k, then we adjoin the packet ofk boxes to the below of
the bottom row ofY_{λ} so that resulting partition is

µ:= (k_{1}, λ_{1}_{i}

1, . . . , λ_{r}_{ir})∈P^{(`)}(n+k).

For example, if we consider the addition of a packet of 5 boxes to the
partitionλ= (32,11)∈P^{(3)}(4), then:

+ =

### BSP in color partitions

Exclusion Rule:

Here index of parts in the partitionλ∈P^{(`)}(n) is important. For any
part ofλ, sayλ_{m}_{im} withi_{m}= 2, we do not allow the addition of a packet
ofk boxes to the row corresponding to the partλ_{m}_{im} inY_{λ}. In short, if
the color of the row corresponding to the part with index 2 is green, we
do not allow the addition of a packet ofk boxes to it.

For instance, forn= 11,`= 3,k = 2 andλ= (62,32,21)∈P^{(3)}(11):

+ =

=

Koustav Banerjee BSP and Partition identities 23 / 45

### BSP in color partitions

Exclusion Rule:

Here index of parts in the partitionλ∈P^{(`)}(n) is important. For any
part ofλ, sayλ_{m}_{im} withi_{m}= 2, we do not allow the addition of a packet
ofk boxes to the row corresponding to the partλ_{m}_{im} inY_{λ}. In short, if
the color of the row corresponding to the part with index 2 is green, we
do not allow the addition of a packet ofk boxes to it.

For instance, forn= 11,`= 3,k = 2 andλ= (62,32,21)∈P^{(3)}(11):

+ =

=

### BSP in color partitions

Exclusion Rule:

Here index of parts in the partitionλ∈P^{(`)}(n) is important. For any
part ofλ, sayλ_{m}_{im} withi_{m}= 2, we do not allow the addition of a packet
ofk boxes to the row corresponding to the partλ_{m}_{im} inY_{λ}. In short, if
the color of the row corresponding to the part with index 2 is green, we
do not allow the addition of a packet ofk boxes to it.

For instance, forn= 11,`= 3,k = 2 andλ= (62,32,21)∈P^{(3)}(11):

+ =

=

Koustav Banerjee BSP and Partition identities 23 / 45

### BSP in color partitions

Exclusion Rule:

^{(`)}(n) is important. For any
part ofλ, sayλ_{m}_{im} withi_{m}= 2, we do not allow the addition of a packet
ofk boxes to the row corresponding to the partλ_{m}_{im} inY_{λ}. In short, if
the color of the row corresponding to the part with index 2 is green, we
do not allow the addition of a packet ofk boxes to it.

For instance, forn= 11,`= 3,k = 2 andλ= (62,32,21)∈P^{(3)}(11):

+ =

=

### BSP in color partitions

Ex 1. We consider all 3 color partitions of 4 and applying the color BSP for adding a packet of 2 boxes to the Young diagram gives:

I. 4_{1}:

+ =

=
II. 3_{1}+ 1_{1}:

+ =

= III. 32+ 11:

+ =

Koustav Banerjee BSP and Partition identities 24 / 45

### BSP in color partitions

Ex 1. We consider all 3 color partitions of 4 and applying the color BSP for adding a packet of 2 boxes to the Young diagram gives:

I. 4_{1}:

+ =

=

II. 3_{1}+ 1_{1}:

+ =

= III. 32+ 11:

+ =

### BSP in color partitions

Ex 1. We consider all 3 color partitions of 4 and applying the color BSP for adding a packet of 2 boxes to the Young diagram gives:

I. 4_{1}:

+ =

=
II. 3_{1}+ 1_{1}:

+ =

=

III. 32+ 11:

+ =

Koustav Banerjee BSP and Partition identities 24 / 45

### BSP in color partitions

I. 4_{1}:

+ =

=
II. 3_{1}+ 1_{1}:

+ =

= III. 32+ 11:

+ =

### BSP in color partitions

IV. 2_{1}+ 2_{1}:

+ =

=

V. 2_{1}+ 1_{1}+ 1_{1}:

+ =

VI. 1_{1}+ 1_{1}+ 1_{1}+ 1_{1}:

+ =

Koustav Banerjee BSP and Partition identities 25 / 45

### BSP in color partitions

IV. 2_{1}+ 2_{1}:

+ =

=

V. 2_{1}+ 1_{1}+ 1_{1}:

+ =

VI. 1_{1}+ 1_{1}+ 1_{1}+ 1_{1}:

+ =

### BSP in color partitions

IV. 2_{1}+ 2_{1}:

+ =

=

V. 2_{1}+ 1_{1}+ 1_{1}:

+ =

VI. 1_{1}+ 1_{1}+ 1_{1}+ 1_{1}:

+ =

Koustav Banerjee BSP and Partition identities 25 / 45

### BSP in color partitions

Ex 2(a). We consider all 3 color partitions of 4 and applying the color BSP for adding a packet of 3 boxes to the Young diagram gives:

I. 4_{1}:

+ =

=
II. 3_{1}+ 1_{1}:

+ =

III. 3_{2}+ 1_{1}:

+ =

### BSP in color partitions

Ex 2(a). We consider all 3 color partitions of 4 and applying the color BSP for adding a packet of 3 boxes to the Young diagram gives:

I. 4_{1}:

+ =

=

II. 3_{1}+ 1_{1}:

+ =

III. 3_{2}+ 1_{1}:

+ =

Koustav Banerjee BSP and Partition identities 26 / 45

### BSP in color partitions

Ex 2(a). We consider all 3 color partitions of 4 and applying the color BSP for adding a packet of 3 boxes to the Young diagram gives:

I. 4_{1}:

+ =

=
II. 3_{1}+ 1_{1}:

+ =

III. 3_{2}+ 1_{1}:

+ =

### BSP in color partitions

I. 4_{1}:

+ =

=
II. 3_{1}+ 1_{1}:

+ =

III. 3_{2}+ 1_{1}:

+ =

Koustav Banerjee BSP and Partition identities 26 / 45

### BSP in color partitions

IV. 2_{1}+ 2_{1}:

+ =

V. 21+ 11+ 11:

+ =

VI. 11+ 11+ 11+ 11:

+ =

### BSP in color partitions

IV. 2_{1}+ 2_{1}:

+ =

V. 21+ 11+ 11:

+ =

VI. 11+ 11+ 11+ 11:

+ =

Koustav Banerjee BSP and Partition identities 27 / 45

### BSP in color partitions

IV. 2_{1}+ 2_{1}:

+ =

V. 21+ 11+ 11:

+ =

VI. 11+ 11+ 11+ 11:

+ =

### BSP in color partitions

Ex 2(b). We consider all 3 color partitions of 4 and applying the color BSP for adding a packet of 3 boxes to the Young diagram gives:

I. 4_{1}:

+ =

II. 3_{1}+ 1_{1}:

+ =

III. 32+ 11:

+ =

=

Koustav Banerjee BSP and Partition identities 28 / 45

### BSP in color partitions

Ex 2(b). We consider all 3 color partitions of 4 and applying the color BSP for adding a packet of 3 boxes to the Young diagram gives:

I. 4_{1}:

+ =

II. 3_{1}+ 1_{1}:

+ =

III. 32+ 11:

+ =

=

### BSP in color partitions

Ex 2(b). We consider all 3 color partitions of 4 and applying the color BSP for adding a packet of 3 boxes to the Young diagram gives:

I. 4_{1}:

+ =

II. 31+ 11:

+ =

III. 32+ 11:

+ =

=

Koustav Banerjee BSP and Partition identities 28 / 45

### BSP in color partitions

I. 4_{1}:

+ =

II. 31+ 11:

+ =

III. 32+ 11:

+ =

=

### BSP in color partitions

IV. 2_{1}+ 2_{1}:

+ =

V. 21+ 11+ 11:

+ =

VI. 1_{1}+ 1_{1}+ 1_{1}+ 1_{1}:

+ =

Koustav Banerjee BSP and Partition identities 29 / 45

### BSP in color partitions

IV. 2_{1}+ 2_{1}:

+ =

V. 21+ 11+ 11:

+ =

VI. 1_{1}+ 1_{1}+ 1_{1}+ 1_{1}:

+ =

### BSP in color partitions

IV. 2_{1}+ 2_{1}:

+ =

V. 21+ 11+ 11:

+ =

VI. 1_{1}+ 1_{1}+ 1_{1}+ 1_{1}:

+ =

Koustav Banerjee BSP and Partition identities 29 / 45