cp's OEIS Frontend

A325552 Number of compositions of n with distinct differences up to sign.

%I A325552 #13 Jan 27 2024 15:26:22
%S A325552 1,1,2,3,6,9,12,23,38,61,78,135,194,315,454,699,982,1495,2102,3085,
%T A325552 4406,6583,9048,13117,18540,26399,36484,51885,72498,100031,139342,
%U A325552 192621,267068,367631,505954,687153,946412,1283367,1745974,2356935,3207554,4311591,5816404
%N A325552 Number of compositions of n with distinct differences up to sign.
%C A325552 A composition of n is a finite sequence of positive integers summing to n.
%C A325552 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 A325552 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 A325552 Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts.</a>
%e A325552 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 A325552 The a(1) = 1 through a(7) = 23 compositions:
%e A325552   (1)  (2)   (3)   (4)    (5)    (6)    (7)
%e A325552        (11)  (12)  (13)   (14)   (15)   (16)
%e A325552              (21)  (22)   (23)   (24)   (25)
%e A325552                    (31)   (32)   (33)   (34)
%e A325552                    (112)  (41)   (42)   (43)
%e A325552                    (211)  (113)  (51)   (52)
%e A325552                           (122)  (114)  (61)
%e A325552                           (221)  (132)  (115)
%e A325552                           (311)  (213)  (124)
%e A325552                                  (231)  (133)
%e A325552                                  (312)  (142)
%e A325552                                  (411)  (214)
%e A325552                                         (223)
%e A325552                                         (241)
%e A325552                                         (322)
%e A325552                                         (331)
%e A325552                                         (412)
%e A325552                                         (421)
%e A325552                                         (511)
%e A325552                                         (1132)
%e A325552                                         (2113)
%e A325552                                         (2311)
%e A325552                                         (3112)
%t A325552 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],UnsameQ@@Abs[Differences[#]]&]],{n,0,15}]
%Y A325552 Cf. A011782, A070211, A175342, A242882, A325325, A325368, A325404, A325545, A325551, A325553, A325555, A325557.
%K A325552 nonn
%O A325552 0,3
%A A325552 _Gus Wiseman_, May 11 2019
%E A325552 a(26)-a(42) from _Alois P. Heinz_, Jan 27 2024