A325358 Number of integer partitions of n whose augmented differences are strictly decreasing.

1, 1, 1, 2, 2, 2, 3, 4, 4, 5, 6, 6, 7, 9, 10, 11, 13, 14, 15, 18, 20, 21, 24, 26, 28, 33, 36, 38, 43, 46, 49, 56, 60, 63, 71, 76, 80, 90, 96, 100, 112, 120, 125, 139, 149, 155, 171, 183, 190, 208, 223, 232, 252, 269, 280, 304, 325, 338, 364, 387, 403

Offset

0,4

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 A325396.

Links

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..1000

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Example

The a(1) = 1 through a(11) = 6 partitions:
  (1)  (2)  (3)   (4)   (5)   (6)   (7)    (8)    (9)    (10)   (11)
            (21)  (31)  (41)  (42)  (52)   (62)   (63)   (73)   (83)
                              (51)  (61)   (71)   (72)   (82)   (92)
                                    (421)  (521)  (81)   (91)   (101)
                                                  (621)  (631)  (731)
                                                         (721)  (821)

Mathematica

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

Crossrefs

Cf. A049988, A240026, A320466, A325349, A325350, A325351, A325356, A325357, A325359, A325393.

Sequence in context: A094838 A025768 A000929 * A029146 A029053 A053254

Adjacent sequences: A325355 A325356 A325357 * A325359 A325360 A325361

Keyword

nonn

Author

Gus Wiseman, May 01 2019

Status

approved