A325324 Number of integer partitions of n whose differences (with the last part taken to be 0) are distinct.

1, 1, 2, 1, 3, 4, 4, 7, 7, 7, 10, 15, 13, 22, 25, 26, 31, 43, 39, 55, 54, 68, 75, 98, 97, 128, 135, 165, 177, 217, 223, 277, 282, 339, 356, 438, 444, 527, 553, 667, 694, 816, 868, 1015, 1054, 1279, 1304, 1538, 1631, 1849, 1958, 2304, 2360, 2701, 2899, 3267

Offset: 0

Author

Gus Wiseman, Apr 23 2019

Keywords

• nonn

Comments

The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (6,3,1) (with the last part taken to be 0) are (-3,-2,-1).

The Heinz numbers of these partitions are given by A325367.

Examples

			The a(1) = 1 through a(11) = 15 partitions (A = 10, B = 11):
  (1)  (2)   (3)  (4)   (5)    (6)    (7)    (8)    (9)    (A)    (B)
       (11)       (22)  (32)   (33)   (43)   (44)   (54)   (55)   (65)
                  (31)  (41)   (51)   (52)   (53)   (72)   (64)   (74)
                        (311)  (411)  (61)   (62)   (81)   (73)   (83)
                                      (322)  (71)   (441)  (82)   (92)
                                      (331)  (332)  (522)  (91)   (A1)
                                      (511)  (611)  (711)  (433)  (443)
                                                           (622)  (533)
                                                           (631)  (551)
                                                           (811)  (632)
                                                                  (641)
                                                                  (722)
                                                                  (731)
                                                                  (911)
                                                                  (6311)
For example, (6,3,1,1) has differences (-3,-2,0,-1), which are distinct, so (6,3,1,1) is counted under a(11).
		

Links

• Fausto A. C. Cariboni, Table of n, a(n) for n = 0..400
• Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Crossrefs

Cf. A007862, A049988, A098859, A130091, A240026, A320348, A320466, A320509, A325325, A325349, A325366, A325367, A325368, A325404, A325407.

Programs

Mathematica

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