A325393 Number of integer partitions of n whose k-th differences are strictly decreasing for all k >= 0.

1, 1, 1, 2, 2, 3, 3, 4, 5, 5, 6, 8, 7, 9, 11, 10, 12, 15, 13, 16, 19, 18, 20, 24, 22, 26, 29, 28, 31, 37, 33, 38, 43, 42, 44, 52, 48, 55, 59, 58, 62, 72, 65, 74, 80, 80, 82, 94, 88, 99, 103, 104, 108, 123, 114, 126, 133, 135, 137, 155, 145, 161, 166, 169, 174

Offset

0,4

Comments

The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (6,3,1) are (-3,-2).

The zeroth differences of a sequence are the sequence itself, while the k-th differences for k > 0 are the differences of the (k-1)-th differences.

The Heinz numbers of these partitions are given by A325399.

Links

Table of n, a(n) for n=0..64.

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

Example

The a(1) = 1 through a(9) = 5 partitions:
  (1)  (2)  (3)   (4)   (5)   (6)   (7)   (8)    (9)
            (21)  (31)  (32)  (42)  (43)  (53)   (54)
                        (41)  (51)  (52)  (62)   (63)
                                    (61)  (71)   (72)
                                          (431)  (81)

Mathematica

Table[Length[Select[IntegerPartitions[n], And@@Table[Greater@@Differences[#, k], {k, 0, Length[#]}]&]], {n, 0, 30}]

Crossrefs

Cf. A049988, A320466, A325353, A325354, A325358, A325391, A325396, A325399, A325404, A325406, A325457, A325468.

Sequence in context: A369113 A241951 A084630 * A320388 A264396 A007360

Adjacent sequences: A325390 A325391 A325392 * A325394 A325395 A325396

Keyword

nonn

Author

Gus Wiseman, May 02 2019

Status

approved