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

Author

Gus Wiseman, Mar 04 2021

Keywords

• nonn

Comments

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

Examples

			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
		

Links

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

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.

Programs

Mathematica

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