Restricted partitions

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Restricted partitions are partitions with conditions on either/both the number of parts or/and the size of the parts.

Restricted partition functions

A restricted partition function gives the number of restricted partitions of $\scriptstyle n\,$, with specific conditions on either/both the number of parts or/and the size of the parts. This is the counterpart of the unrestricted partition function, namely the partition function, which gives the number of unrestricted partitions, namely partitions, of $\scriptstyle n\,$.

Restricted partitions

Restricted size of parts

Parts restricted to equal size

For $\scriptstyle n\,>\,0\,$, restricted partitions with parts of equal size correspond to divisors of n, the number of restricted partitions with parts of equal size being the number of divisors of n.

Parts restricted to equal size

$n\,$	Partitions	Number of partitions
0	{ { } }	1
1	{ {1} }	1
2	{ {1, 1}, {2} }	2
3	{ {1, 1, 1}, {3} }	2
4	{ {1, 1, 1, 1}, {2, 2}, {4} }	3
5	{ {1, 1, 1, 1, 1}, {5} }	2
6	{ {1, 1, 1, 1, 1, 1}, {2, 2, 2}, {3, 3}, {6} }	4
7	{ {1, 1, 1, 1, 1, 1, 1}, {7} }	2
8	{ {1, 1, 1, 1, 1, 1, 1, 1}, {2, 2, 2, 2}, {4, 4}, {8} }	4
9	{ {1, 1, 1, 1, 1, 1, 1, 1, 1}, {3, 3, 3}, {9} }	3
10	{ {1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, {2, 2, 2, 2, 2}, {5, 5}, {10} }	4
11	{ {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, {11} }	2
12	{ {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, {2, 2, 2, 2, 2, 2}, {3, 3, 3, 3}, {4, 4, 4}, {6, 6}, {12} }	6

Number of partitions of $\scriptstyle n\,$ into parts of equal size, $\scriptstyle n\,\geq \,0\,$. Number of divisors of $\scriptstyle n,\,n\,\geq \,1\,$. (Cf. A000005)

{1, 1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6, 2, 4, 4, 5, 2, 6, 2, 6, 4, 4, 2, 8, 3, 4, 4, 6, 2, 8, 2, 6, 4, 4, 4, 9, 2, 4, 4, 8, 2, 8, 2, 6, 6, 4, 2, 10, 3, 6, 4, 6, 2, 8, 4, 8, 4, 4, 2, 12, 2, 4, 6, 7, 4, ...}

Parts restricted to Fibonacci numbers

Parts restricted to Fibonacci numbers (with a single type of 1)

Parts restricted to Fibonacci numbers (with a single type of 1)

$n\,$	Partitions	Number of partitions
0	{ { } }	1
1	{ {1} }	1
2	{ {1, 1}, {2} }	2
3	{ {1, 1, 1}, {1, 2}, {3} }	3
4	{ {1, 1, 1, 1}, {1, 1, 2}, {1, 3}, {2, 2} }	4
5	{ {1, 1, 1, 1, 1}, {1, 1, 1, 2}, {1, 1, 3}, {1, 2, 2}, {2, 3}, {5} }	6
6	{ {1, 1, 1, 1, 1, 1}, {1, 1, 1, 1, 2}, {1, 1, 1, 3}, {1, 1, 2, 2}, {1, 2, 3}, {1, 5}, {2, 2, 2}, {3, 3} }	8
7	{ {1, 1, 1, 1, 1, 1, 1}, {1, 1, 1, 1, 1, 2}, {1, 1, 1, 1, 3}, {1, 1, 1, 2, 2}, {1, 1, 2, 3}, {1, 1, 5}, {1, 2, 2, 2}, {1, 3, 3}, {2, 2, 3}, {2, 5} }	10
8	{ {1, 1, 1, 1, 1, 1, 1, 1}, {1, 1, 1, 1, 1, 1, 2}, {1, 1, 1, 1, 1, 3}, {1, 1, 1, 1, 2, 2}, {1, 1, 1, 2, 3}, {1, 1, 1, 5}, {1, 1, 2, 2, 2}, {1, 1, 3, 3}, {1, 2, 2, 3}, {1, 2, 5}, {2, 2, 2, 2}, {2, 3, 3}, {3, 5}, {8} }	14
9	{ {1, 1, 1, 1, 1, 1, 1, 1, 1}, ... }	17
10	{ {1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, ... }	22
11	{ {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, ... }	27
12	{ {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, ... }	33

Number of partitions of $\scriptstyle n\,$ into Fibonacci parts (with a single type of 1), $\scriptstyle n\,\geq \,0\,$. (Cf. A003107)

{1, 1, 2, 3, 4, 6, 8, 10, 14, 17, 22, 27, 33, 41, 49, 59, 71, 83, 99, 115, 134, 157, 180, 208, 239, 272, 312, 353, 400, 453, 509, 573, 642, 717, 803, 892, 993, 1102, 1219, 1350, ...}

Parts restricted to Fibonacci numbers (with 2 types of 1)

Parts restricted to Fibonacci numbers (with 2 types of 1,) 1 being $\scriptstyle F_{1}\,$, 1 being $\scriptstyle F_{2}\,$.

Parts restricted to Fibonacci numbers (with 2 types of 1)

$n\,$	Partitions	Number of partitions
0	{ { } }	1
1	{ {1}, {1} }	2
2	{ {1, 1}, {1, 1}, {1, 1}, {2} }	4
3	{ {1, 1, 1}, {1, 1, 1}, {1, 1, 1}, {1, 2}, {1, 1, 1}, {1, 2}, {3} }	7
4	{ {1, 1, 1, 1}, {1, 1, 1, 1}, {1, 1, 1, 1}, {1, 1, 2}, {1, 1, 1, 1}, {1, 1, 2}, {1, 3}, {1, 1, 1, 1}, {1, 1, 2}, {1, 3}, {2, 2} }	11
5	{ {1, 1, 1, 1, 1}, ... }	17
6	{ {1, 1, 1, 1, 1, 1}, ... }	25
7	{ {1, 1, 1, 1, 1, 1, 1}, ... }	35
8	{ {1, 1, 1, 1, 1, 1, 1, 1}, ... }	14
9	{ {1, 1, 1, 1, 1, 1, 1, 1, 1}, ... }	49
10	{ {1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, ... }	66
11	{ {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, ... }	88
12	{ {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, ... }	115

Number of partitions of $\scriptstyle n\,$ into Fibonacci parts (with 2 types of 1), $\scriptstyle n\,\geq \,0\,$. (Cf. A007000)

{1, 2, 4, 7, 11, 17, 25, 35, 49, 66, 88, 115, 148, 189, 238, 297, 368, 451, 550, 665, 799, 956, 1136, 1344, 1583, 1855, 2167, 2520, 2920, 3373, 3882, 4455, 5097, 5814, 6617, ...}

Parts restricted to nonconsecutive Fibonacci numbers

Parts restricted to nonconsecutive Fibonacci numbers

$n\,$	Partitions	Number of partitions
0	{ { } }	1
1	{ {1} }	1
2	{ {2} }	1
3	{ {1, 2} }	1
4	{ {1, 3} }	1
5	{ {5} }	1
6	{ {1, 5} }	1
7	{ {2, 5} }	1
8	{ {8} }	1
9	{ {1, 8} }	1
10	{ {2, 8} }	1
11	{ {3, 8} }	1
12	{ {1, 3, 8} }	1

Number of partitions of $\scriptstyle n\,$ into nonconsecutive Fibonacci parts (with either type of 1), $\scriptstyle n\,\geq \,0\,$. (Cf. A000012)

{1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...}

Parts restricted to powers

Parts restricted to powers of 2

Parts restricted to powers of 2

$n\,$	Partitions	Number of partitions
0	{ { } }	1
1	{ {1} }	1
2	{ {1, 1}, {2} }	2
3	{ {1, 1, 1}, {1, 2} }	2
4	{ {1, 1, 1, 1}, {1, 1, 2}, {2, 2}, {4} }	4
5	{ {1, 1, 1, 1, 1}, {1, 1, 1, 2}, {1, 2, 2}, {1, 4} }	4
6	{ {1, 1, 1, 1, 1, 1}, {1, 1, 1, 1, 2}, {1, 1, 2, 2}, {1, 1, 4}, {2, 2, 2}, {2, 4} }	6
7	{ {1, 1, 1, 1, 1, 1, 1}, {1, 1, 1, 1, 1, 2}, {1, 1, 1, 2, 2}, {1, 1, 1, 4}, {1, 2, 2, 2}, {1, 2, 4} }	6
8	{ {1, 1, 1, 1, 1, 1, 1, 1}, {1, 1, 1, 1, 1, 1, 2}, {1, 1, 1, 1, 2, 2}, {1, 1, 1, 1, 4}, {1, 1, 2, 2, 2}, {1, 1, 2, 4}, {2, 2, 2, 2}, {2, 2, 4}, {4, 4}, {8} }	10
9	{ {1, 1, 1, 1, 1, 1, 1, 1, 1}, ... }	10
10	{ {1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, ... }	14
11	{ {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, ... }	14
12	{ {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, ... }	20

Binary partition function: number of partitions of $\scriptstyle n\,$ into powers of 2, $\scriptstyle n\,\geq \,0\,$. (Cf. A018819)

{1, 1, 2, 2, 4, 4, 6, 6, 10, 10, 14, 14, 20, 20, 26, 26, 36, 36, 46, 46, 60, 60, 74, 74, 94, 94, 114, 114, 140, 140, 166, 166, 202, 202, 238, 238, 284, 284, 330, 330, 390, 390, 450, ...}

Parts restricted to powers of 3

Parts restricted to powers of 3

$n\,$	Partitions	Number of partitions
0	{ { } }	1
1	{ {1} }	1
2	{ {1, 1} }	1
3	{ {1, 1, 1}, {3} }	2
4	{ {1, 1, 1, 1}, {1, 3} }	2
5	{ {1, 1, 1, 1, 1}, {1, 1, 3} }	2
6	{ {1, 1, 1, 1, 1, 1}, {1, 1, 1, 3}, {3, 3} }	3
7	{ {1, 1, 1, 1, 1, 1, 1}, {1, 1, 1, 1, 3}, {1, 3, 3} }	3
8	{ {1, 1, 1, 1, 1, 1, 1, 1}, {1, 1, 1, 1, 1, 3}, {1, 1, 3, 3} }	3
9	{ {1, 1, 1, 1, 1, 1, 1, 1, 1}, {1, 1, 1, 1, 1, 1, 3}, {1, 1, 1, 3, 3}, {3, 3, 3}, {9} }	5
10	{ {1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, ... }	5
11	{ {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, ... }	5
12	{ {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}, ... }	7

Ternary partition function: number of partitions of $\scriptstyle n\,$ into powers of 3, $\scriptstyle n\,\geq \,0\,$. (Cf. A062051)

{1, 1, 1, 2, 2, 2, 3, 3, 3, 5, 5, 5, 7, 7, 7, 9, 9, 9, 12, 12, 12, 15, 15, 15, 18, 18, 18, 23, 23, 23, 28, 28, 28, 33, 33, 33, 40, 40, 40, 47, 47, 47, 54, 54, 54, 63, 63, 63, 72, 72, ...}

Parts restricted to unit, primes, nonprimes, composites or noncomposites

Parts restricted to noncomposites (unit or primes)

Number of partitions of $\scriptstyle n\,$ into unit or prime, i.e. noncomposite, parts, $\scriptstyle n\,\geq \,1\,$; number of different products of partitions of $\scriptstyle n\,$; number of distinct orders of Abelian subgroups of symmetric group S_n. (Cf. A034891)

{1, 2, 3, 4, 6, 8, 11, 14, 18, 23, 29, 36, 45, 55, 67, 81, 98, 117, 140, 166, 196, 231, 271, 317, 369, 429, 496, 573, 660, 758, 869, 993, 1133, 1290, 1465, 1662, 1881, 2125, 2397, ...}

Parts restricted to primes

Parts restricted to primes

$n\,$	Partitions	Number of partitions
0	{ { } }	1
1	{ }	0
2	{ {2} }	1
3	{ {3} }	1
4	{ {2, 2} }	1
5	{ {2, 3}, {5} }	2
6	{ {2, 2, 2}, {3, 3} }	2
7	{ {2, 2, 3}, {2, 5}, {7} }	3
8	{ {2, 2, 2, 2}, {2, 3, 3}, {3, 5} }	3
9	{ {2, 2, 2, 3}, {2, 2, 5}, {2, 7}, {3, 3, 3} }	4
10	{ {2, 2, 2, 2, 2}, {2, 2, 3, 3}, {2, 3, 5}, {3, 7}, {5, 5} }	5
11	{ {2, 2, 2, 2, 3}, {2, 2, 2, 5}, {2, 2, 7}, {2, 3, 3, 3}, {3, 3, 5}, {11} }	6
12	{ {2, 2, 2, 2, 2, 2}, {2, 2, 2, 3, 3}, {2, 2, 3, 5}, {2, 3, 7}, {2, 5, 5}, {3, 3, 3, 3}, {5, 7} }	7

Number of partitions of $\scriptstyle n\,$ into prime parts, $\scriptstyle n\,\geq \,0\,$. (Cf. A000607)

{1, 0, 1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 9, 10, 12, 14, 17, 19, 23, 26, 30, 35, 40, 46, 52, 60, 67, 77, 87, 98, 111, 124, 140, 157, 175, 197, 219, 244, 272, 302, 336, 372, 413, 456, ...}

Parts restricted to nonprimes (unit or composites)

Number of partitions of $\scriptstyle n\,$ into unit or composite, i.e. nonprime, parts, $\scriptstyle n\,\geq \,0\,$. (Cf. A002095)

{1, 1, 1, 1, 2, 2, 3, 3, 5, 6, 8, 8, 12, 13, 17, 19, 26, 28, 37, 40, 52, 58, 73, 79, 102, 113, 139, 154, 191, 210, 258, 284, 345, 384, 462, 509, 614, 679, 805, 893, 1060, 1171, 1382, ...}

Parts restricted to composites

Number of partitions of $\scriptstyle n\,$ into composite, $\scriptstyle n\,\geq \,0\,$. (Cf. A023895)

{1, 0, 0, 0, 1, 0, 1, 0, 2, 1, 2, 0, 4, 1, 4, 2, 7, 2, 9, 3, 12, 6, 15, 6, 23, 11, 26, 15, 37, 19, 48, 26, 61, 39, 78, 47, 105, 65, 126, 88, 167, 111, 211, 146, 264, 196, 331, 241, 426, ...}

Restricted number of parts

A026820(n,k) gives the number of partitions of n with at most k parts.

Restricted number of parts and restricted size of parts

Restricted number of parts, parts restricted to powers

Parts restricted to at most one of any powers of 2

Number of partitions of $\scriptstyle n\,$ into at most 1 of any powers of 2, $\scriptstyle n\,\geq \,0\,$. (uniqueness of base 2 representation) (Cf. A000012)

{1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...}

Parts restricted to at most k-1 of any powers of k

Number of partitions of $\scriptstyle n\,$ into at most $\scriptstyle k-1\,$ of any powers of k, $\scriptstyle k\,>\,1,\,n\,\geq \,0\,$. (uniqueness of base k representation) (Cf. A000012)

{1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...}

Restricted number of parts, parts restricted to primes

Parts restricted to two primes for even numbers and three primes for odd numbers

According to Goldbach conjecture, every even integer $\scriptstyle n\,\geq \,4\,=\,2+2\,$ is the sum of two primes in at least one way, and every odd integer $\scriptstyle n\,\geq \,7\,=\,2+2+3\,$ is the sum of three primes in at least one way.

Parts restricted to two primes for even numbers and three primes for odd numbers

$n\,$	Partitions	Number of partitions
0	{ }	0
1	{ }	0
2	{ }	0
3	{ }	0
4	{ {2, 2} }	1
5	{ }	0
6	{ {3, 3} }	1
7	{ {2, 2, 3} }	1
8	{ {3, 5} }	1
9	{ {2, 2, 5}, {3, 3, 3} }	2
10	{ {3, 7}, {5, 5} }	2
11	{ {2, 2, 7}, {3, 3, 5} }	2
12	{ {5, 7} }	1
13	{ {3, 3, 7}, {3, 5, 5} }	2
14	{ {3, 11}, {7, 7} }	2
15	{ {2, 2, 11}, {3, 5, 7}, {5, 5, 5} }	3
16	{ {3, 13}, {5, 11} }	2
17	{ {2, 2, 13}, {3, 3, 11}, {3, 7, 7}, {5, 5, 7} }	4
18	{ {5, 13}, {7, 11} }	2
19	{ {3, 3, 13}, {3, 5, 11}, {5, 7, 7} }	3
20	{ {3, 17}, {7, 13} }	2
21	{ {2, 2, 17}, {3, 5, 13}, {3, 7, 11}, {5, 5, 11}, {7, 7, 7} }	5
22	{ {3, 19}, {5, 17}, {11, 11} }	3
23	{ {2, 2, 19}, {3, 3, 17}, {3, 7, 13}, {5, 5, 13}, {5, 7, 11} }	5
24	{ {5, 19}, {7, 17}, {11, 13} }	3
25	{ {3, 3, 19}, {3, 5, 17}, {3, 11, 11}, {5, 7, 13}, {7, 7, 11} }	5

Number of partitions of odd numbers into three primes and of even numbers into two primes. (Cf. A083338$\scriptstyle (n),\,n\,\geq \,1\,$)

{0, 0, 0, 1, 0, 1, 1, 1, 2, 2, 2, 1, 2, 2, 3, 2, 4, 2, 3, 2, 5, 3, 5, 3, 5, 3, 7, 2, 7, 3, 6, 2, 9, 4, 8, 4, 9, 2, 10, 3, 11, 4, 10, 3, 12, 4, 13, 5, 12, 4, 15, 3, 16, 5, 14, 3, 17, 4, 16, 6, 16, ...}

Parts restricted to two odd primes for even numbers and three odd primes for odd numbers

According to Goldbach conjecture, every even integer $\scriptstyle n\,\geq \,6\,=\,3+3\,$ is the sum of two odd primes in at least one way, and every odd integer $\scriptstyle n\,\geq \,9\,=\,3+3+3\,$ is the sum of three odd primes in at least one way.

Parts restricted to two odd primes for even numbers and three odd primes for odd numbers

$n\,$	Partitions	Number of partitions
0	{ }	0
1	{ }	0
2	{ }	0
3	{ }	0
4	{ }	0
5	{ }	0
6	{ {3, 3} }	1
7	{ }	0
8	{ {3, 5} }	1
9	{ {3, 3, 3} }	1
10	{ {3, 7}, {5, 5} }	2
11	{ {3, 3, 5} }	1
12	{ {5, 7} }	1
13	{ {3, 3, 7}, {3, 5, 5} }	2
14	{ {3, 11}, {7, 7} }	2
15	{ {3, 5, 7}, {5, 5, 5} }	2
16	{ {3, 13}, {5, 11} }	2
17	{ {3, 3, 11}, {3, 7, 7}, {5, 5, 7} }	3
18	{ {5, 13}, {7, 11} }	2
19	{ {3, 3, 13}, {3, 5, 11}, {5, 7, 7} }	3
20	{ {3, 17}, {7, 13} }	2
21	{ {3, 5, 13}, {3, 7, 11}, {5, 5, 11}, {7, 7, 7} }	4
22	{ {3, 19}, {5, 17}, {11, 11} }	3
23	{ {3, 3, 17}, {3, 7, 13}, {5, 5, 13}, {5, 7, 11} }	4
24	{ {5, 19}, {7, 17}, {11, 13} }	3
25	{ {3, 3, 19}, {3, 5, 17}, {3, 11, 11}, {5, 7, 13}, {7, 7, 11} }	5

Number of partitions of odd numbers into three odd primes and of even numbers into two odd primes. (Cf. A??????$\scriptstyle (n),\,n\,\geq \,1\,$)

{0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 1, 1, 2, 2, 2, 2, 3, 2, 3, 2, 4, 3, 4, 3, 5, 3, 6, 2, 7, 3, 6, 2, 8, 4, 7, 4, 9, 2, 10, 3, 10, 4, 10, 3, 11, 4, 12, 5, 12, 4, 14, 3, 16, 5, 14, 3, 16, 4, 16, 6, 16, 3, ...}

From Goldbach strong conjecture: number of decompositions of $\scriptstyle 2n\,$ into an unordered sum of two odd primes. (Cf. A002375$\scriptstyle (n),\,n\,\geq \,1\,$)

{0, 0, 1, 1, 2, 1, 2, 2, 2, 2, 3, 3, 3, 2, 3, 2, 4, 4, 2, 3, 4, 3, 4, 5, 4, 3, 5, 3, 4, 6, 3, 5, 6, 2, 5, 6, 5, 5, 7, 4, 5, 8, 5, 4, 9, 4, 5, 7, 3, 6, 8, 5, 6, 8, 6, 7, 10, 6, 6, 12, 4, 5, 10, 3, 7, ...}

From Goldbach weak conjecture: number of decompositions of $\scriptstyle 2n+1\,$ into an unordered sum of three odd primes. (Cf. A007963$\scriptstyle (n),\,n\,\geq \,0\,$)

{0, 0, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 6, 7, 6, 8, 7, 9, 10, 10, 10, 11, 12, 12, 14, 16, 14, 16, 16, 16, 18, 20, 20, 20, 21, 21, 21, 27, 24, 25, 28, 27, 28, 33, 29, 32, 35, 34, 30, 37, 36, ...}

See also

• 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 $\scriptstyle \Omega (n)\,$, the number of prime factors (with repetition) of $\scriptstyle n\,$.)

• Ordered prime signature (which may be viewed as a given composition of $\scriptstyle \Omega (n)\,$, the number of prime factors (with repetition) of $\scriptstyle n\,$.)

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

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