A325357 Number of integer partitions of n whose augmented differences are strictly increasing.

1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 3, 5, 5, 4, 5, 6, 5, 7, 7, 7, 7, 9, 7, 10, 10, 8, 11, 13, 10, 13, 14, 12, 14, 17, 13, 17, 19, 17, 18, 22, 19, 22, 24, 21, 24, 28, 24, 29, 30, 28, 31, 35, 30, 35, 40, 36

Offset

0,5

Comments

The augmented differences aug(y) of an integer partition y of length k are given by aug(y)_i = y_i - y_{i + 1} + 1 if i < k and aug(y)_k = y_k. For example, aug(6,5,5,3,3,3) = (2,1,3,1,1,3).

The Heinz numbers of these partitions are given by A325395.

Links

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..2000

Example

The a(28) = 10 partitions:
  (28)
  (18,10)
  (17,11)
  (16,12)
  (15,13)
  (14,14)
  (12,10,6)
  (11,10,7)
  (10,10,8)
  (8,8,7,5)
For example, the augmented differences of (8,8,7,5) are (1,2,3,5), which are strictly increasing.

Mathematica

aug[y_]:=Table[If[i<Length[y], y[[i]]-y[[i+1]]+1, y[[i]]], {i, Length[y]}];
Table[Length[Select[IntegerPartitions[n], Less@@aug[#]&]], {n, 0, 30}]

Crossrefs

Cf. A000837, A007294, A049988, A098859, A325351, A325356, A325360, A325391, A325395.

Sequence in context: A045917 A240708 A235645 * A294107 A240874 A029379

Adjacent sequences: A325354 A325355 A325356 * A325358 A325359 A325360

Keyword

nonn

Author

Gus Wiseman, Apr 23 2019

Status

approved