A350844 Number of strict integer partitions of n with no difference -2.

1, 1, 1, 2, 1, 3, 3, 4, 4, 7, 7, 8, 11, 12, 15, 18, 21, 23, 31, 32, 40, 45, 54, 59, 73, 78, 94, 106, 122, 136, 161, 177, 203, 231, 259, 293, 334, 372, 417, 476, 525, 592, 663, 742, 821, 931, 1020, 1147, 1271, 1416, 1558, 1752, 1916, 2137, 2357, 2613, 2867

Offset

0,4

Links

Table of n, a(n) for n=0..56.

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

Example

The a(1) = 1 through a(12) = 11 partitions (A..C = 10..12):
  1   2   3    4   5    6     7    8     9     A      B     C
          21       32   51    43   62    54    73     65    84
                   41   321   52   71    63    82     74    93
                              61   521   72    91     83    A2
                                         81    541    92    B1
                                         432   721    A1    543
                                         621   4321   632   651
                                                      821   732
                                                            741
                                                            921
                                                            6321

Mathematica

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

Crossrefs

The version for no difference 0 is A000009.

The version for no difference > -2 is A001227, non-strict A034296.

The version for no difference -1 is A003114 (A325160).

The version for subsets of prescribed maximum is A005314.

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

The opposite version is A072670.

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

The non-strict version is A350842.

A000041 counts integer partitions.

A027187 counts partitions of even length.

A027193 counts partitions of odd length (A026424).

A116931 counts partitions with no difference -1 (A319630).

A323092 counts double-free integer partitions (A320340) strict A120641.

A325534 counts separable partitions (A335433).

A325535 counts inseparable partitions (A335448).

Cf. A000929, A003000, A018819, A040039, A045690, A045691, A154402, A303362, A323094, A342095, A342097.

Sequence in context: A027157 A112194 A238788 * A083041 A318611 A366472

Adjacent sequences: A350841 A350842 A350843 * A350845 A350846 A350847

Keyword

nonn

Author

Gus Wiseman, Jan 21 2022

Status

approved