A325552 - OEIS

A325552

Number of compositions of n with distinct differences up to sign.

%I #13 Jan 27 2024 15:26:22

%S 1,1,2,3,6,9,12,23,38,61,78,135,194,315,454,699,982,1495,2102,3085,

%T 4406,6583,9048,13117,18540,26399,36484,51885,72498,100031,139342,

%U 192621,267068,367631,505954,687153,946412,1283367,1745974,2356935,3207554,4311591,5816404

%N Number of compositions of n with distinct differences up to sign.

%C A composition of n is a finite sequence of positive integers summing to n.

%C The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (3,1,2) are (-2,1).

%C a(n) has the same parity as n for n > 0, since reversing a composition does not change whether or not it has this property, and the only valid symmetric compositions are (n) and (n/2,n/2), with the latter only existing for even n. - _Charlie Neder_, Jun 06 2019

%H Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts.</a>

%e The differences of (1,2,1) are (1,-1), which are different but not up to sign, so (1,2,1) is not counted under a(4).

%e The a(1) = 1 through a(7) = 23 compositions:

%e (1) (2) (3) (4) (5) (6) (7)

%e (11) (12) (13) (14) (15) (16)

%e (21) (22) (23) (24) (25)

%e (31) (32) (33) (34)

%e (112) (41) (42) (43)

%e (211) (113) (51) (52)

%e (122) (114) (61)

%e (221) (132) (115)

%e (311) (213) (124)

%e (231) (133)

%e (312) (142)

%e (411) (214)

%e (223)

%e (241)

%e (322)

%e (331)

%e (412)

%e (421)

%e (511)

%e (1132)

%e (2113)

%e (2311)

%e (3112)

%t Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],UnsameQ@@Abs[Differences[#]]&]],{n,0,15}]

%Y Cf. A011782, A070211, A175342, A242882, A325325, A325368, A325404, A325545, A325551, A325553, A325555, A325557.

%K nonn

%O 0,3

%A _Gus Wiseman_, May 11 2019

%E a(26)-a(42) from _Alois P. Heinz_, Jan 27 2024