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A325324 Number of integer partitions of n whose differences (with the last part taken to be 0) are distinct. 17
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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.

LINKS

Table of n, a(n) for n=0..55.

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

EXAMPLE

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).

MATHEMATICA

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

CROSSREFS

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

Sequence in context: A248393 A033779 A033799 * A241745 A210040 A285329

Adjacent sequences:  A325321 A325322 A325323 * A325325 A325326 A325327

KEYWORD

nonn

AUTHOR

Gus Wiseman, Apr 23 2019

STATUS

approved

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Last modified January 21 05:23 EST 2020. Contains 331104 sequences. (Running on oeis4.)