Restricted compositions

This article page is a stub, please help by expanding it.

Restricted size of parts

(...)

Restricted number of parts

Note that since the empty sum is defined as 0, the number of compositions into 0 parts of 0 is 1.

The number of compositions into

k

parts of a positive integer

n

is (choose

k  −  1

separators among the

n  −  1

potential separators between

n

stones in a single row)
     

c (n, k )  =  ( n − 1 k − 1 ), 1 ≤ k ≤ n.

The number of compositions with at most

k

parts of a positive integer

n

is
     

k ∑   i  = 1    c  (n, i )  =    k ∑   i  = 1    ( n − 1 i − 1 ), 1 ≤ k ≤ n.

The number of compositions with at least

k

parts of a positive integer

n

is
     

n ∑   i  = k    c  (n, i )  =    n ∑   i  = k    ( n − 1 i − 1 ), 1 ≤ k ≤ n.

The number of compositions with an odd number of parts of a positive integer

n

is

(...)

The number of compositions with an even number of parts of a positive integer

n

is

(...)

Restricted number of parts and restricted size of parts

n parts of size at most m

Compositions of

n   ≤   k   ≤   m n

in to

n

parts, with size of parts not greater than

m

.
The number of compositions of

n   ≤   k   ≤   m n

into

n

parts with size of parts not greater than

m

is given by the coefficient of

x k

in

(  ∑ m i  = 1 x i  )  n = x n (  ∑ m − 1 i  = 0 x i  )  n

, thus by the [univariate]

m

-nomial coefficient

Cm(n ,   j )

of

x   j = x k  − n

in

(  ∑ m − 1 i  = 0 x i  )  n

.
The [univariate]

m

-nomial coefficients,

m   ≥   2

, are obtained by the recurrence relation

Cm(0, 0)  :=  1;

Cm(n ,   j )  :=    j ∑   i  = j − (m − 1) n    Cm(n − 1, i ), n ≥ 1, 0 ≤   j ≤ (m − 1) n,

where

Cm(n,   j )   :=   0

for

j < 0

or

j > (m  −  1) n

.
For [univariate]

m

-nomial coefficient array, row

n   ≥   0

is the sequence of coefficients of

x   j, 0   ≤    j   ≤   (m  −  1) n ,

in

(  ∑ m  − 1 i  = 0 x i  )  n

.

m	[Univariate] m -nomial coefficients (concatenated rows, row n   ≥   0  )	A-number
2	1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 5, 10, 10, 5, 1, 1, 6, 15, 20, 15, 6, 1, 1, 7, 21, 35, 35, 21, 7, 1, 1, 8, 28, 56, 70, 56, 28, 8, 1, 1, 9, 36, 84, 126, 126, 84, 36, 9, 1, 1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1, 1, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1, ...	A007318
3	1, 1, 1, 1, 1, 2, 3, 2, 1, 1, 3, 6, 7, 6, 3, 1, 1, 4, 10, 16, 19, 16, 10, 4, 1, 1, 5, 15, 30, 45, 51, 45, 30, 15, 5, 1, 1, 6, 21, 50, 90, 126, 141, 126, 90, 50, 21, 6, 1, 1, 7, 28, 77, 161, 266, 357, 393, 357, 266, 161, 77, 28, 7, 1, 1, 8, 36, 112, 266, ...	A027907
4	1, 1, 1, 1, 1, 1, 2, 3, 4, 3, 2, 1, 1, 3, 6, 10, 12, 12, 10, 6, 3, 1, 1, 4, 10, 20, 31, 40, 44, 40, 31, 20, 10, 4, 1, 1, 5, 15, 35, 65, 101, 135, 155, 155, 135, 101, 65, 35, 15, 5, 1, 1, 6, 21, 56, 120, 216, 336, 456, 546, 580, 546, 456, 336, 216, 120, 56, 21, 6, 1, ...	A008287
5	1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 4, 3, 2, 1, 1, 3, 6, 10, 15, 18, 19, 18, 15, 10, 6, 3, 1, 1, 4, 10, 20, 35, 52, 68, 80, 85, 80, 68, 52, 35, 20, 10, 4, 1, 1, 5, 15, 35, 70, 121, 185, 255, 320, 365, 381, 365, 320, 255, 185, 121, 70, 35, 15, 5, 1, 1, 6, 21, 56, 126, 246, 426, 666, ...	A035343
6	1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1, 1, 3, 6, 10, 15, 21, 25, 27, 27, 25, 21, 15, 10, 6, 3, 1, 1, 4, 10, 20, 35, 56, 80, 104, 125, 140, 146, 140, 125, 104, 80, 56, 35, 20, 10, 4, 1, ...	A063260
7	1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 6, 5, 4, 3, 2, 1, 1, 3, 6, 10, 15, 21, 28, 33, 36, 37, 36, 33, 28, 21, 15, 10, 6, 3, 1, 1, 4, 10, 20, 35, 56, 84, 116, 149, 180, 206, 224, 231, 224, 206, 180, 149, 116, 84, 56, 35, ...	A063265
8	1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 3, 2, 1, 1, 3, 6, 10, 15, 21, 28, 36, 42, 46, 48, 48, 46, 42, 36, 28, 21, 15, 10, 6, 3, 1, 1, 4, 10, 20, 35, 56, 84, 120, 161, 204, 246, 284, 315, 336, 344, 336, 315, 284, 246, 204, 161, 120, 84, 56, 35, 20, 10, 4, 1, ...	A171890
9	1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 8, 7, 6, 5, 4, 3, 2, 1, 1, 3, 6, 10, 15, 21, 28, 36, 45, 52, 57, 60, 61, 60, 57, 52, 45, 36, 28, 21, 15, 10, 6, 3, 1, 1, 4, 10, 20, 35, 56, 84, 120, 165, 216, 270, 324, 375, 420, 456, 480, 489, 480, 456, ...	A213652
10	1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 63, 69, 73, 75, 75, 73, 69, 63, 55, 45, 36, 28, 21, 15, 10, 6, 3, 1, 1, 4, 10, 20, 35, 56, 84, 120, 165, 220, 282, 348, 415, 480, ...	A213651
11	1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 75, 82, 87, 90, 91, 90, 87, 82, 75, 66, 55, 45, 36, 28, 21, 15, 10, 6, 3, 1, ...	A??????
12	1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 88, 96, 102, 106, 108, 108, 106, 102, 96, 88, 78, 66, 55, 45, 36, 28, 21, 15, 10, 6, 3, 1, ...	A??????

Table of possible outcomes
2 throws of “

m

-sided” dice (or 2 spins of

m

-slotted roulette)
All

k

th antidiagonal pairs,

1   ≤   k   ≤   2 m  −  1

, sum to

k + 1

	1	2	...	m − 1	m
1	(1, 1)	(1, 2)	(1, ...)	(1, m − 1)	(1, m)
2	(2, 1)	(2, 2)	(2, ...)	(2, m − 1)	(2, m)
...	(..., 1)	(..., 2)	(..., ...)	(..., m − 1)	(..., m)
m − 1	(m − 1, 1)	(m − 1, 2)	(m − 1, ...)	(m − 1, m − 1)	(m − 1, m)
m	(m, 1)	(m, 2)	(m, ...)	(m, m − 1)	(m, m)

Coins

Coins (e.g., considering head as 1 and tail as 2) may be taken as “dihedral dice.”

“Dihedral dice” (

n

coins,

n   ≥   2

): compositions of

n   ≤   k   ≤   2 n

into

n

parts, with size of parts not greater than 2.
The number of compositions of

n   ≤   k   ≤   2 n

into

n

parts with size of parts not greater than 2 is given by the (binomials being either [univariate] or [bivariate], since

n2 = n  −  n1

) binomial coefficient

(  nk  )

of

x k

in

(x 1 + x 2 ) n = x n (x 0 + x 1 ) n

.

Number of compositions of

n   ≤   k   ≤   2 n

with

n

“dihedral dice”
Row

n, n   ≥   0,

is the sequence of [univariate] binomial coefficients of

x   j, 0   ≤   j   ≤   n ,

in

(x 0 + x 1 ) n

n	Number of compositions	A-number
0	1	A??????
1	1, 1,	A??????
2	1, 2, 1	A??????
3	1, 3, 3, 1	A??????
4	1, 4, 6, 4, 1	A??????
5	1, 5, 10, 10, 5, 1	A??????
6	1, 6, 15, 20, 15, 6, 1	A??????
7	1, 7, 21, 35, 35, 21, 7, 1	A??????

A007318 Pascal’s triangle read by rows:

C (n, k ) = (  nk  )  = n! k ! (n  −  k )!  , 0   ≤   k   ≤   n

.

{1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 5, 10, 10, 5, 1, 1, 6, 15, 20, 15, 6, 1, 1, 7, 21, 35, 35, 21, 7, 1, 1, 8, 28, 56, 70, 56, 28, 8, 1, 1, 9, 36, 84, 126, 126, 84, 36, 9, 1, 1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1, 1, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1, ...}

Dice

Tetrahedral dice

Tetrahedral dice (

n

dice,

n   ≥   2

): compositions of

n   ≤   k   ≤   4 n

into

n

parts, with size of parts not greater than 4.
The number of compositions of

n   ≤   k   ≤   4 n

into

n

parts with size of parts not greater than 4 is given by the coefficient of

x k

in

(x 1 + x 2 + x 3 + x 4 ) n = x n (x 0 + x 1 + x 2 + x 3 ) n

.

Number of compositions of

n   ≤   k   ≤   4 n

with

n

tetrahedral dice
Row

n, n   ≥   0,

is the sequence of [univariate] quadrinomial coefficients of

x   j, 0   ≤   j   ≤   3 n ,

in

(x 0 + x 1 + x 2 + x 3 ) n

n	Number of compositions	A-number
0	1	A??????
1	1, 1, 1, 1	A??????
2	1, 2, 3, 4, 3, 2, 1	A??????
3	1, 3, 6, 10, 12, 12, 10, 6, 3, 1	A??????
4	1, 4, 10, 20, 31, 40, 44, 40, 31, 20, 10, 4, 1	A??????
5	1, 5, 15, 35, 65, 101, 135, 155, 155, 135, 101, 65, 35, 15, 5, 1	A??????
6	1, 6, 21, 56, 120, 216, 336, 456, 546, 580, 546, 456, 336, 216, 120, 56, 21, 6, 1	A??????
7	1, 7, 28, 84, 203, 413, 728, 1128, 1554, 1918, 2128, 2128, 1918, 1554, 1128, 728, 413, 203, 84, 28, 7, 1	A??????

A008287 Triangle of [univariate] quadrinomial (also called tetranomial) coefficients. Row

n, n   ≥   0,

is the sequence of coefficients of

(1 + x + x 2 + x 3 ) n

.

{1, 1, 1, 1, 1, 1, 2, 3, 4, 3, 2, 1, 1, 3, 6, 10, 12, 12, 10, 6, 3, 1, 1, 4, 10, 20, 31, 40, 44, 40, 31, 20, 10, 4, 1, 1, 5, 15, 35, 65, 101, 135, 155, 155, 135, 101, 65, 35, 15, 5, 1, 1, 6, 21, 56, 120, 216, 336, 456, 546, 580, 546, 456, 336, 216, 120, 56, 21, 6, 1, ...}

Cubic dice

Cubic dice (

n

dice,

n   ≥   2

): compositions of

n   ≤   k   ≤   6 n

into

n

parts, with size of parts not greater than 6.
The number of compositions of

n   ≤   k   ≤   6 n

into

n

parts with size of parts not greater than 6 is given by the coefficient of

x k

in

(x 1 + x 2 + x 3 + x 4 + x 5 + x 6 ) n = x n (x 0 + x 1 + x 2 + x 3 + x 4 + x 5 ) n

.

Number of compositions of

n   ≤   k   ≤   6 n

with

n

cubic dice
Row

n, n   ≥   0,

is the sequence of [univariate] hexanomial coefficients of

x   j, 0   ≤   j   ≤   5 n ,

in

(x 0 + x 1 + x 2 + x 3 + x 4 + x 5 ) n

n	Number of compositions	A-number
0	1
1	1, 1, 1, 1, 1, 1
2	1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1	A??????
3	1, 3, 6, 10, 15, 21, 25, 27, 27, 25, 21, 15, 10, 6, 3, 1	A056150
4	1, 4, 10, 20, 35, 56, 80, 104, 125, 140, 146, 140, 125, 104, 80, 56, 35, 20, 10, 4, 1	A??????
5	1, 5, 15, 35, 70, 126, 205, 305, 420, 540, 651, 735, 780, 780, 735, 651, 540, 420, 305, 205, 126, 70, 35, 15, 5, 1	A??????
6	1, 6, 21, 56, 126, 252, 456, 756, 1161, 1666, 2247, 2856, 3431, 3906, 4221, 4332, 4221, 3906, 3431, 2856, 2247, 1666, 1161, 756, 456, 252, 126, 56, 21, 6, 1	A108907
7	1, 7, 28, 84, 210, 462, 917, 1667, 2807, 4417, 6538, 9142, 12117, 15267, 18327, 20993, 22967, 24017, 24017, 22967, 20993, 18327, 15267, 12117, 9142, 6538, 4417, 2807, 1667, 917, 462, 210, 84, 28, 7, 1	A166322

A063260 [Univariate] sextinomial (also called hexanomial) coefficient array. Row

n, n   ≥   0,

is the sequence of coefficients of

(  ∑ 5 i  = 0 x i  )  n

.

{1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1, 1, 3, 6, 10, 15, 21, 25, 27, 27, 25, 21, 15, 10, 6, 3, 1, 1, 4, 10, 20, 35, 56, 80, 104, 125, 140, 146, 140, 125, 104, 80, 56, 35, 20, 10, 4, 1, ...}

Octahedral dice

Octahedral dice (

n

dice,

n   ≥   2

): compositions of

n   ≤   k   ≤   8 n

into

n

parts, with size of parts not greater than 8.
The number of compositions of

n   ≤   k   ≤   8 n

into

n

parts with size of parts not greater than 8 is given by the coefficient of

x k

in

(x 1 + x 2 + x 3 + x 4 + x 5 + x 6 + x 7 + x 8 ) n = x n (x 0 + x 1 + x 2 + x 3 + x 4 + x 5 + x 6 + x 7 ) n

.

Number of compositions of

n   ≤   k   ≤   8 n

with

n

octahedral dice
Row

n, n   ≥   0,

is the sequence of [univariate] octanomial coefficients of

x   j, 0   ≤   j   ≤   7n ,

in

(x 0 + x 1 + x 2 + x 3 + x 4 + x 5 + x 6 + x 7 ) n

n	Number of compositions	A-number
0	1	A??????
1	1, 1, 1, 1, 1, 1, 1, 1	A??????
2	1, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 3, 2, 1	A??????
3	1, 3, 6, 10, 15, 21, 28, 36, 42, 46, 48, 48, 46, 42, 36, 28, 21, 15, 10, 6, 3, 1	A??????
4	1, 4, 10, 20, 35, 56, 84, 120, 161, 204, 246, 284, 315, 336, 344, 336, 315, 284, 246, 204, 161, 120, 84, 56, 35, 20, 10, 4, 1	A??????
5	1, 5, 15, 35, 70, 126, 210, 330, 490, 690, 926, 1190, 1470, 1750, 2010, 2226, 2380, 2460, 2460, 2380, 2226, 2010, 1750, 1470, 1190, 926, 690, 490, 330, 210, 126, 70, 35, 15, 5, 1	A??????

A171890 [Univariate] octonomial coefficient array. Row

n, n   ≥   0,

is the sequence of coefficients of

(  ∑ 7 i  = 0 x i  )  n

.

{1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 7, 6, 5, 4, 3, 2, 1, 1, 3, 6, 10, 15, 21, 28, 36, 42, 46, 48, 48, 46, 42, 36, 28, 21, 15, 10, 6, 3, 1, 1, 4, 10, 20, 35, 56, 84, 120, 161, 204, 246, 284, 315, 336, 344, 336, 315, 284, 246, 204, 161, 120, 84, 56, 35, 20, 10, 4, 1, ...}

Dodecahedral dice

Dodecahedral dice (

n

dice,

n   ≥   2

): compositions of

n   ≤   k   ≤   12 n

into

n

parts, with size of parts not greater than 12.
The number of compositions of

n   ≤   k   ≤   12 n

into

n

parts with size of parts not greater than 12 is given by the coefficient of

x k

in

(x 1 + x 2 + x 3 + x 4 + x 5 + x 6 + x 7 + x 8 + x 9 + x 10 + x 11 + x 12 ) n = x n (x 0 + x 1 + x 2 + x 3 + x 4 + x 5 + x 6 + x 7 + x 8 + x 9 + x 10 + x 11 ) n

.

Number of compositions of

n   ≤   k   ≤   12 n

with

n

dodecahedral dice
Row

n, n   ≥   0,

is the sequence of [univariate] dodecanomial coefficients of

x   j, 0   ≤   j   ≤   11 n ,

in

(x 0 + x 1 + x 2 + x 3 + x 4 + x 5 + x 6 + x 7 + x 8 + x 9 + x 10 + x 11 ) n

n	Number of compositions	A-number
0	1	A??????
1	1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1	A??????
2	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1	A??????
3	1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 88, 96, 102, 106, 108, 108, 106, 102, 96, 88, 78, 66, 55, 45, 36, 28, 21, 15, 10, 6, 3, 1	A??????
4	1, 4, 10, 20, 35, 56, 84, 120, 165, 220, 286, 364, 451, 544, 640, 736, 829, 916, 994, 1060, 1111, 1144, 1156, 1144, 1111, 1060, 994, 916, 829, 736, 640, 544, 451, 364, 286, 220, 165, 120, 84, 56, 35, 20, 10, 4, 1	A??????

A?????? [Univariate] dodecanomial coefficient array. Row

n, n   ≥   0,

is the sequence of coefficients of

(  ∑ 11 i  = 0 x i  )  n

.

{1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 88, 96, 102, 106, 108, 108, 106, 102, 96, 88, 78, 66, 55, 45, 36, 28, 21, 15, 10, 6, 3, 1, ...}

Icosahedral dice

Icosahedral dice (

n

dice,

n   ≥   2

): compositions of

n   ≤   k   ≤   20 n

into

n

parts, with size of parts not greater than 20.
The number of compositions of

n   ≤   k   ≤   20 n

into

n

parts with size of parts not greater than 20 is given by the coefficient of

x k

in

(  ∑ 20 i  = 1 x i  )  n = x n (  ∑ 19 i  = 0 x i  )  n

.

Number of compositions of

n   ≤   k   ≤   20 n

with

n

icosahedral dice
Row

n, n   ≥   0,

is the sequence of [univariate] icosanomial coefficients of

x   j, 0   ≤   j   ≤   19 n ,

in

(  ∑ 19 i  = 0 x i  )  n

n	Number of compositions	A-number
0	1	A??????
1	1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1	A??????
2	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1	A??????
3	1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 228, 244, 258, 270, 280, 288, 294, 298, 300, 300, 298, 294, 288, 280, 270, 258, 244, 228, 210, 190, 171, 153, 136, 120, 105, 91, 78, 66, 55, 45, 36, 28, 21, 15, 10, 6, 3, 1	A??????

A?????? [Univariate] icosanomial coefficient array. Row

n, n   ≥   0,

is the sequence of coefficients of

(  ∑ 19 i  = 0 x i  )  n

.

{1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, ...}

See also

• Signature permutations for bijections of compositions

• Unordered partitions
  • Partitions and restricted partitions
  • Partition function and restricted partition functions
  • Partition identities
  • Orderings of partitions
  • Multidimensional partitions
    • Partitions (one-dimensional partitions)
    • Plane partitions (two-dimensional partitions)
    • Solid partitions (three-dimensional partitions)

• Ordered partitions
  • Compositions and restricted compositions
  • Composition function and restricted composition functions
  • Composition identities
  • Orderings of compositions
  • Multidimensional compositions
    • Compositions (one-dimensional ordered partitions)
    • Plane compositions (two-dimensional ordered partitions)
    • Solid compositions (three-dimensional ordered partitions)

• Multiplicative partitions (unordered factorizations)
  • Factorizations and restricted factorizations
  • Factorization function and restricted factorization functions

• Multiplicative compositions (ordered factorizations)
  • Ordered factorizations and restricted ordered factorizations
  • Ordered factorization function and restricted ordered factorization functions

• Prime signature (which may be viewed as a given partition of

  Ω (n)

  , the number of prime factors (with repetition) of

  n

  )

• Ordered prime signature (which may be viewed as a given composition of

  Ω (n)

  , the number of prime factors (with repetition) of

  n

  )

• Sets
  • Set partitions
  • Restricted set partitions
  • Set compositions
  • Restricted set compositions

• Multisets
  • Multiset partitions
  • Restricted multiset partitions
  • Multiset compositions
  • Restricted multiset compositions

External links

• Gašper Jaklič, Vito Vitrih and Emil Žagar, Closed Form Formula for the Number of Restricted Compositions.
• Weisstein, Eric W., Multinomial Coefficient, from MathWorld—A Wolfram Web Resource.