A350842 Number of integer partitions of n with no difference -2.

1, 1, 2, 3, 4, 6, 9, 12, 16, 24, 30, 40, 54, 69, 89, 118, 146, 187, 239, 297, 372, 468, 575, 711, 880, 1075, 1314, 1610, 1947, 2359, 2864, 3438, 4135, 4973, 5936, 7090, 8466, 10044, 11922, 14144, 16698, 19704, 23249, 27306, 32071, 37639, 44019, 51457, 60113

Offset

0,3

Links

Table of n, a(n) for n=0..48.

Gus Wiseman, Sequences counting and ranking partitions and compositions by their differences and quotients.

Example

The a(1) = 1 through a(7) = 12 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (21)   (22)    (32)     (33)      (43)
             (111)  (211)   (41)     (51)      (52)
                    (1111)  (221)    (222)     (61)
                            (2111)   (321)     (322)
                            (11111)  (411)     (511)
                                     (2211)    (2221)
                                     (21111)   (3211)
                                     (111111)  (4111)
                                               (22111)
                                               (211111)
                                               (1111111)

Mathematica

Table[Length[Select[IntegerPartitions[n], FreeQ[Differences[#], -2]&]], {n, 0, 30}]

Crossrefs

Heinz number rankings are in parentheses below.

The version for no difference 0 is A000009.

The version for subsets of prescribed maximum is A005314.

The version for all differences < -2 is A025157, non-strict A116932.

The version for all differences > -2 is A034296, strict A001227.

The opposite version is A072670.

The version for no difference -1 is A116931 (A319630), strict A003114.

The multiplicative version is A350837 (A350838), strict A350840.

The strict case is A350844.

The complement for quotients is counted by A350846 (A350845).

A000041 = integer partitions.

A027187 = partitions of even length.

A027193 = partitions of odd length (A026424).

A323092 = double-free partitions (A320340), strict A120641.

A325534 = separable partitions (A335433).

A325535 = inseparable partitions (A335448).

A350839 = partitions with a gap and conjugate gap (A350841).

Cf. A000070, A000929, A001511, A003242, A007359, A018819, A040039, A045690, A045691, A101417, A154402, A323093.

Sequence in context: A286141 A007604 A013950 * A018550 A248475 A035952

Adjacent sequences: A350839 A350840 A350841 * A350843 A350844 A350845

Keyword

nonn

Author

Gus Wiseman, Jan 20 2022

Status

approved