A325875 Number of compositions of n whose differences of all degrees > 1 are nonzero.

1, 1, 2, 3, 7, 13, 20, 38, 69, 129, 222, 407, 726, 1313, 2318, 4146, 7432, 13296, 23759, 42458, 75714

Offset: 0

Author

Gus Wiseman, Jun 02 2019

Keywords

• nonn
• more

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. If m is the length of the sequence, its differences of all degrees are the union of the zeroth through m-th differences.

A composition of n is a finite sequence of positive integers with sum n.

The case for all degrees including 1 is A325851.

Examples

			The a(1) = 1 through a(6) = 20 compositions:
  (1)  (2)   (3)   (4)    (5)     (6)
       (11)  (12)  (13)   (14)    (15)
             (21)  (22)   (23)    (24)
                   (31)   (32)    (33)
                   (112)  (41)    (42)
                   (121)  (113)   (51)
                   (211)  (122)   (114)
                          (131)   (132)
                          (212)   (141)
                          (221)   (213)
                          (311)   (231)
                          (1121)  (312)
                          (1211)  (411)
                                  (1122)
                                  (1131)
                                  (1212)
                                  (1311)
                                  (2121)
                                  (2211)
                                  (11211)
		

Links

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

Crossrefs

Cf. A049988, A238423, A325325, A325468, A325545, A325849, A325850, A325851, A325852, A325874, A325876.

Programs

Mathematica

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!MemberQ[Union@@Table[Differences[#,i],{i,2,Length[#]}],0]&]],{n,0,10}]