A325852 Number of (strict) integer partitions of n whose differences of all degrees are nonzero.

1, 1, 1, 2, 2, 3, 3, 5, 6, 6, 9, 11, 11, 15, 19, 19, 26, 31, 31, 41, 49, 53, 62, 75, 81, 97, 112, 124, 145, 171, 175, 215, 244, 274, 307, 344, 388, 446, 497, 561, 599, 700, 779, 881, 981, 1054, 1184, 1340, 1500, 1669, 1767, 2031, 2237, 2486, 2765, 2946, 3300

Offset: 0

Author

Gus Wiseman, May 31 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) are (-3,-2). The zeroth differences are the sequence itself, while k-th differences for k > 0 are the differences of the (k-1)-th differences. The differences of all degrees of a sequence are the union of its zeroth through m-th differences, where m is the length of the sequence.

Examples

			The a(1) = 1 through a(11) = 11 partitions (A = 10, B = 11):
  (1)  (2)  (3)   (4)   (5)   (6)   (7)    (8)    (9)    (A)    (B)
            (21)  (31)  (32)  (42)  (43)   (53)   (54)   (64)   (65)
                        (41)  (51)  (52)   (62)   (63)   (73)   (74)
                                    (61)   (71)   (72)   (82)   (83)
                                    (421)  (431)  (81)   (91)   (92)
                                           (521)  (621)  (532)  (A1)
                                                         (541)  (542)
                                                         (631)  (632)
                                                         (721)  (641)
                                                                (731)
                                                                (821)
		

Links

• Fausto A. C. Cariboni, Table of n, a(n) for n = 0..250

Crossrefs

The case for only degrees > 1 is A325874.

Cf. A049988, A175342, A238423, A279945, A295370, A325328, A325468, A325545, A325850, A325851, A325875.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],!MemberQ[Union@@Table[Differences[#,i],{i,Length[#]}],0]&]],{n,0,30}]