A364675 Number of integer partitions of n whose nonzero first differences are a submultiset of the parts.

1, 1, 2, 3, 4, 4, 7, 7, 10, 12, 15, 15, 26, 25, 35, 45, 55, 60, 86, 94, 126, 150, 186, 216, 288, 328, 407, 493, 610, 699, 896, 1030, 1269, 1500, 1816, 2130, 2620, 3029, 3654, 4300, 5165, 5984, 7222, 8368, 9976, 11637, 13771, 15960, 18978, 21896, 25815, 29915

Offset

0,3

Comments

Conjecture: For subsets of {1..n} instead of partitions of n we have A101925.

Conjecture: The strict version is A154402.

Links

Table of n, a(n) for n=0..51.

Example

The partition y = (3,2,1,1) has first differences (1,1,0), and (1,1) is a submultiset of y, so y is counted under a(7).
The a(1) = 1 through a(8) = 10 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (221)    (33)      (421)      (44)
             (111)  (211)   (2111)   (42)      (2221)     (422)
                    (1111)  (11111)  (222)     (3211)     (2222)
                                     (2211)    (22111)    (4211)
                                     (21111)   (211111)   (22211)
                                     (111111)  (1111111)  (32111)
                                                          (221111)
                                                          (2111111)
                                                          (11111111)

Mathematica

submultQ[cap_, fat_] := And@@Function[i, Count[fat, i] >= Count[cap, i]] /@ Union[List@@cap];
Table[Length[Select[IntegerPartitions[n], submultQ[Differences[Union[#]], #]&]], {n, 0, 30}]

Crossrefs

For subsets of {1..n} we appear to have A101925, A364671, A364672.

The strict case (no differences of 0) appears to be A154402.

Starting with the distinct parts gives A342337.

For disjoint multisets: A363260, subsets A364463, strict A364464.

For overlapping multisets: A364467, ranks A364537, strict A364536.

For subsets instead of submultisets we have A364673.

A000041 counts integer partitions, strict A000009.

A008284 counts partitions by length, strict A008289.

A236912 counts sum-free partitions, complement A237113.

A325325 counts partitions with distinct first differences.

Cf. A002865, A007862, A108917, A229816, A237667, A237668, A320347, A363225, A364272, A364345, A364466.

Sequence in context: A342337 A167932 A006087 * A241315 A136330 A294267

Adjacent sequences: A364672 A364673 A364674 * A364676 A364677 A364678

Keyword

nonn

Author

Gus Wiseman, Aug 04 2023

Status

approved