A325553 Number of compositions of n with distinct circular differences up to sign.

1, 1, 1, 1, 1, 1, 1, 7, 21, 31, 41, 87, 99, 191, 245, 381, 501, 735, 883, 1309, 1841, 2589, 3435, 4941, 6857, 9791, 13503, 19475, 27073, 37175, 52299, 72249, 100359, 139317, 190549, 256769, 355193, 471963, 644433, 858793, 1159161, 1530879, 2056073, 2711921

Offset: 0

Author

Gus Wiseman, May 11 2019

Keywords

• nonn

Comments

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

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).

Examples

			The a(1) = 1 through a(8) = 21 compositions:
  (1)  (2)  (3)  (4)  (5)  (6)  (7)    (8)
                                (124)  (125)
                                (142)  (134)
                                (214)  (143)
                                (241)  (152)
                                (412)  (215)
                                (421)  (251)
                                       (314)
                                       (341)
                                       (413)
                                       (431)
                                       (512)
                                       (521)
                                       (1124)
                                       (1142)
                                       (1241)
                                       (1421)
                                       (2114)
                                       (2411)
                                       (4112)
                                       (4211)
		

Links

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

Crossrefs

Cf. A000079, A008965, A167606, A173258, A325324, A325349, A325545, A325549, A325551, A325552, A325553, A325556, A325558.

Programs

Mathematica

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],UnsameQ@@Abs[Differences[Append[#,First[#]]]]&]],{n,20}]

Extensions

a(0) and a(26)-a(43) from Alois P. Heinz, Jan 28 2024