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A325358 Number of integer partitions of n whose augmented differences are strictly decreasing. 9
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 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=0..60.

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

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Last modified April 3 04:21 EDT 2020. Contains 333195 sequences. (Running on oeis4.)