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

1,2

Comments

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

Links

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

Example

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

Mathematica

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

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.

Sequence in context: A026799 A185326 A238209 * A210716 A027190 A036824

Adjacent sequences: A342093 A342094 A342095 * A342097 A342098 A342099

Keyword

nonn

Author

Gus Wiseman, Mar 02 2021

Status

approved