A317081 Number of integer partitions of n whose multiplicities cover an initial interval of positive integers.

1, 1, 1, 2, 3, 5, 5, 9, 11, 16, 20, 30, 34, 50, 58, 79, 96, 129, 152, 203, 243, 307, 375, 474, 563, 707, 850, 1042, 1246, 1532, 1815, 2215, 2632, 3173, 3765, 4525, 5323, 6375, 7519, 8916, 10478, 12414, 14523, 17133, 20034, 23488, 27422, 32090, 37285, 43511, 50559

Offset: 0

Author

Gus Wiseman, Jul 21 2018

Keywords

• nonn

Comments

Also the number of integer partitions of n with distinct section-sums, where the k-th part of the section-sum partition is the sum of all (distinct) parts that appear at least k times. - Gus Wiseman, Apr 21 2025

Examples

			The a(1) = 1 through a(9) = 16 partitions:
 (1) (2) (3)  (4)   (5)   (6)   (7)    (8)    (9)
         (21) (31)  (32)  (42)  (43)   (53)   (54)
              (211) (41)  (51)  (52)   (62)   (63)
                    (221) (321) (61)   (71)   (72)
                    (311) (411) (322)  (332)  (81)
                                (331)  (422)  (432)
                                (421)  (431)  (441)
                                (511)  (521)  (522)
                                (3211) (611)  (531)
                                       (3221) (621)
                                       (4211) (711)
                                              (3321)
                                              (4221)
                                              (4311)
                                              (5211)
                                              (32211)
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 0..160

Crossrefs

The case with parts also covering an initial interval is A317088.

These partitions are ranked by A317090.

A000041 counts integer partitions, strict A000009.

A008284 counts partitions by length, strict A008289.

A047966 counts partitions with constant section-sums.

A048767 interchanges prime indices and prime multiplicities (Look-and-Say), see A048768.

A055932 lists numbers whose prime indices cover an initial interval.

A116540 counts normal set multipartitions.

A304442 counts partitions with equal run-sums, ranks A353833.

A381436 lists the section-sum partition of prime indices.

A381440 lists the Look-and-Say partition of prime indices.

Cf. A000837, A069799, A217605, A317082, A317084, A317085, A317087.

Cf. A003242, A116861, A239455, A353837, A364916, A381431, A381432, A381438.

Programs

Mathematica

normalQ[m_]:=Union[m]==Range[Max[m]];
Table[Length[Select[IntegerPartitions[n],normalQ[Length/@Split[#]]&]],{n,30}]

Python

from sympy.utilities.iterables import partitions
def A317081(n):
    if n == 0:
        return 1
    c = 0
    for d in partitions(n):
        s = set(d.values())
        if len(s) == max(s):
            c += 1
    return c # Chai Wah Wu, Jun 22 2020