A342096 Number of integer partitions of n with no adjacent parts having quotient >= 2.

1, 2, 2, 3, 3, 4, 4, 6, 6, 8, 9, 11, 13, 17, 19, 24, 29, 35, 42, 51, 61, 75, 90, 108, 130, 158, 189, 227, 272, 325, 389, 464, 553, 659, 782, 929, 1102, 1306, 1545, 1824, 2153, 2538, 2989, 3514, 4127, 4842, 5673, 6642, 7766, 9068, 10583, 12335, 14361, 16705

Offset: 1

Author

Gus Wiseman, Mar 02 2021

Keywords

• nonn

Comments

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

Examples

			The a(1) = 1 through a(10) = 8 partitions:
  1  2   3    4     5      6       7        8         9          A
     11  111  22    32     33      43       44        54         55
              1111  11111  222     322      53        333        64
                           111111  1111111  332       432        433
                                            2222      3222       532
                                            11111111  111111111  3322
                                                                 22222
                                                                 1111111111
		

Links

• Fausto A. C. Cariboni, Table of n, a(n) for n = 1..250

Crossrefs

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

The multiplicative version is A342083 or A342084.

The version allowing quotients of 2 exactly is A342094.

The strict case allowing quotients of 2 exactly is A342095.

The strict case is A342097.

The reciprocal version is A342098.

A000009 counts strict partitions.

A000929 counts partitions with no adjacent parts having quotient < 2.

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, A342085, A342191.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],And@@Thread[Differences[-#]