A368557 Number of compositions of n such that the set of absolute differences is a subset of the set of parts.

1, 1, 1, 3, 2, 2, 11, 10, 13, 27, 58, 87, 157, 253, 438, 850, 1462, 2474, 4472, 7716, 13544, 24115, 42360, 74013, 131038, 229009, 401946, 707293, 1242059, 2177682, 3828831, 6716062, 11777179, 20678592, 36267148, 63586772, 111556751, 195610763, 342949281

Offset: 0

Author

John Tyler Rascoe, Dec 29 2023

Keywords

• nonn

Examples

			For n=12, composition [2,1,2,4,3] of 12 has the set of absolute differences {1,2}, which is a subset of the set of parts {1,2,3,4}, so it counts towards a(12) = 157.
a(3) = 3 compositions: [3], [2,1], [1,2].
a(6) = 11 compositions: [6], [4,2], [2,4], [3,2,1], [3,1,2], [2,3,1], [2,1,3], [1,3,2], [1,2,3], [2,1,2,1], [1,2,1,2].
		

Links

• John Tyler Rascoe, Python program

Crossrefs

Cf. A003242, A032020, A173258, A214248, A214270.

Programs

Mathematica

g[0] = {{}}; g[n_Integer] := g[n] = Flatten[ParallelTable[Append[#, i] & /@ g[n - i], {i, 1, n}], 1];
isC[p_List] := Module[{d}, d = Abs[Differences[p]]; Union[d] === Union[Select[d, MemberQ[p, #] &]]];
a[n_Integer] := a[n] = Count[g[n], p_ /; isC[p]];
Monitor[Table[a[n], {n, 0, 19}], {n, Table[a[m], {m, 0, n - 1}]}] (* Robert P. P. McKone, Jan 02 2024 *)

Extensions

a(24)-a(38) from Alois P. Heinz, Dec 30 2023