login
Number of compositions of n with distinct circular differences.
5

%I #6 May 11 2019 18:31:53

%S 1,1,3,6,11,8,26,50,79,121,195,265,478,742,1269,1914,2929,4462,6825,

%T 10309,16324,24633,37213,56828,84482

%N Number of compositions of n with distinct circular differences.

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

%C The circular differences of a composition c of length k are c_{i + 1} - c_i for i < k and c_1 - c_i for i = k. For example, the circular differences of (1,2,1,3) are (1,-1,2,-2), which are distinct, so (1,2,1,3) is counted under a(7).

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

%e The a(1) = 1 through a(7) = 26 compositions:

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

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

%e (21) (31) (23) (24) (25)

%e (112) (32) (42) (34)

%e (121) (41) (51) (43)

%e (211) (113) (114) (52)

%e (122) (141) (61)

%e (131) (411) (115)

%e (212) (124)

%e (221) (133)

%e (311) (142)

%e (151)

%e (214)

%e (223)

%e (232)

%e (241)

%e (313)

%e (322)

%e (331)

%e (412)

%e (421)

%e (511)

%e (1213)

%e (1312)

%e (2131)

%e (3121)

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

%Y Cf. A000079, A000740, A008965, A242882, A325545, A325549, A325553, A325558, A325589, A325591.

%K nonn,more

%O 1,3

%A _Gus Wiseman_, May 10 2019