A349430 Number of set partitions of [5n] into 5-element subsets {i, i+k, i+2k, i+3k, i+4k} with 1<=k<=n.

1, 1, 2, 4, 10, 20, 58, 124, 344, 811, 2071, 4973, 15454, 36031, 96212, 237563, 668695, 1626751, 4674373, 11470722, 31460456, 81705943, 224598113

Offset: 0

Author

Alois P. Heinz, Nov 17 2021

Keywords

• nonn
• more

Examples

			a(4) = 10: {{1,2,3,4,5}, {6,7,8,9,10}, {11,12,13,14,15}, {16,17,18,19,20}},
  {{1,3,5,7,9}, {2,4,6,8,10}, {11,12,13,14,15}, {16,17,18,19,20}},
  {{1,2,3,4,5}, {6,8,10,12,14}, {7,9,11,13,15}, {16,17,18,19,20}},
  {{1,4,7,10,13}, {2,5,8,11,14}, {3,6,9,12,15}, {16,17,18,19,20}},
  {{1,2,3,4,5}, {6,7,8,9,10}, {11,13,15,17,19}, {12,14,16,18,20}},
  {{1,3,5,7,9}, {2,4,6,8,10}, {11,13,15,17,19}, {12,14,16,18,20}},
  {{1,5,9,13,17}, {2,4,6,8,10}, {3,7,11,15,19}, {12,14,16,18,20}},
  {{1,2,3,4,5}, {6,9,12,15,18}, {7,10,13,16,19}, {8,11,14,17,20}},
  {{1,3,5,7,9}, {2,6,10,14,18}, {4,8,12,16,20}, {11,13,15,17,19}},
  {{1,5,9,13,17}, {2,6,10,14,18}, {3,7,11,15,19}, {4,8,12,16,20}}.
		

Links

• Wikipedia, Partition of a set

Crossrefs

Cf. A000567 (number of subsets), A008587 (number of elements), A104431 (when k is unbounded), A337520.

Main diagonal of A360491.

Programs

Maple

b:= proc(s, t) option remember; `if`(s={}, 1, (m-> add(
     `if`({seq(m-h*j, h=1..4)} minus s={}, b(s minus {seq(m-h*j,
      h=0..4)}, t), 0), j=1..min(t, iquo(m-1, 4))))(max(s)))
    end:
a:= proc(n) option remember; forget(b): b({$1..5*n}, n) end:
seq(a(n), n=0..10);

Mathematica

b[s_, t_] := b[s, t] = If[s == {}, 1, Function[m, Sum[If[Union[Table[m - h*j, {h, 1, 4}] ~Complement~ s] == {}, b[s  ~Complement~ Union[Table[m - h*j, {h, 0, 4}]], t], 0], {j, 1, Min[t, Quotient[m-1, 4]]}]][Max[s]]];
a[n_] := a[n] = b[Range[5n], n];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 15}] (* Jean-François Alcover, May 16 2022, after Alois P. Heinz *)

Extensions

a(22) from Alois P. Heinz, Nov 23 2022