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

0,8

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).

Links

Table of n, a(n) for n=0..43.

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

Example

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)

Mathematica

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

Crossrefs

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

Sequence in context: A160890 A319527 A297178 * A032639 A264619 A015729

Adjacent sequences: A325550 A325551 A325552 * A325554 A325555 A325556

Keyword

nonn

Author

Gus Wiseman, May 11 2019

Extensions

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

Status

approved