A325468 Number of integer partitions y of n such that the k-th differences of y are distinct (independently) for all k >= 0.

1, 1, 1, 2, 2, 3, 3, 5, 6, 6, 9, 11, 10, 15, 17, 19, 24, 31, 26, 40, 43, 51, 52, 72, 66, 89, 88, 111, 119, 150, 130, 183, 193, 229, 231, 279, 287, 358, 365, 430, 426, 538, 535, 649, 680, 742, 803, 943, 982, 1136, 1115

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 A325467.

Links

Table of n, a(n) for n=0..50.

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

Example

The a(1) = 1 through a(9) = 6 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)
                                    (421)  (431)  (81)
                                           (521)  (621)

Mathematica

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

Crossrefs

Cf. A000009, A325324, A325325, A325349, A325353, A325354, A325391, A325393, A325404, A325406, A325467.

Sequence in context: A345162 A316313 A325876 * A320347 A178932 A325852

Adjacent sequences: A325465 A325466 A325467 * A325469 A325470 A325471

Keyword

nonn

Author

Gus Wiseman, May 03 2019

Status

approved