A342098 Number of (necessarily strict) integer partitions of n with all adjacent parts having quotients > 2.

1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 5, 5, 6, 7, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 21, 23, 25, 26, 28, 31, 33, 35, 38, 40, 42, 45, 48, 51, 55, 58, 61, 65, 68, 72, 77, 81, 85, 90, 94, 98, 104, 109, 114, 121, 127, 132, 139, 146

Offset

1,4

Comments

The decapitation of such a partition (delete the greatest part) is term-wise less than its negated first-differences.

Links

Fausto A. C. Cariboni, Table of n, a(n) for n = 1..3000

Example

The a(1) = 1 through a(16) = 8 partitions (A..G = 10..16):
  1  2  3  4   5   6   7   8   9   A   B    C    D    E    F    G
           31  41  51  52  62  72  73  83   93   94   A4   B4   B5
                       61  71  81  82  92   A2   A3   B3   C3   C4
                                   91  A1   B1   B2   C2   D2   D3
                                       731  831  C1   D1   E1   E2
                                                 931  941  A41  F1
                                                      A31  B31  B41
                                                                C31

Mathematica

Table[Length[Select[IntegerPartitions[n], And@@Thread[Differences[-#]>Rest[#]]&]], {n, 30}]

Crossrefs

The version allowing equality is A000929.

The case of equality (all adjacent parts having quotient 2) is A154402.

The multiplicative version is A342083.

The version with all quotients <= 2 is A342094 (strict: A342095).

The version with all quotients < 2 is A342096 (strict: A342097).

A000009 counts strict partitions.

A003114 counts partitions with adjacent parts differing by more than 1.

A034296 counts partitions with adjacent parts differing by at most 1.

Cf. A027193, A001055, A001227, A003242, A167606, A178470, A337135, A342084, A342085.

Sequence in context: A265410 A029249 A025770 * A283875 A099773 A140471

Adjacent sequences: A342095 A342096 A342097 * A342099 A342100 A342101

Keyword

nonn

Author

Gus Wiseman, Mar 04 2021

Status

approved