A274537 Number T(n,k) of set partitions of [n] into k blocks such that each element is contained in a block whose index parity coincides with the parity of the element; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 3, 2, 1, 0, 0, 1, 3, 7, 2, 1, 0, 0, 1, 7, 14, 13, 3, 1, 0, 0, 1, 7, 35, 26, 22, 3, 1, 0, 0, 1, 15, 70, 113, 66, 34, 4, 1, 0, 0, 1, 15, 155, 226, 311, 102, 50, 4, 1, 0, 0, 1, 31, 310, 833, 933, 719, 200, 70, 5, 1
Offset: 0
Examples
T(6,2) = 1: 135|246. T(6,3) = 3: 13|246|5, 15|246|3, 1|246|35. T(6,4) = 7: 13|24|5|6, 15|24|3|6, 1|24|35|6, 15|26|3|4, 15|2|3|46, 1|26|35|4, 1|2|35|46. T(6,5) = 2: 1|26|3|4|5, 1|2|3|46|5. T(6,6) = 1: 1|2|3|4|5|6. Triangle T(n,k) begins: 1; 0, 1; 0, 0, 1; 0, 0, 1, 1; 0, 0, 1, 1, 1; 0, 0, 1, 3, 2, 1; 0, 0, 1, 3, 7, 2, 1; 0, 0, 1, 7, 14, 13, 3, 1; 0, 0, 1, 7, 35, 26, 22, 3, 1; 0, 0, 1, 15, 70, 113, 66, 34, 4, 1; 0, 0, 1, 15, 155, 226, 311, 102, 50, 4, 1; ...
Links
- Alois P. Heinz, Rows n = 0..140, flattened
- Wikipedia, Partition of a set
Crossrefs
Programs
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Maple
b:= proc(n, m, t) option remember; `if`(n=0, x^m, add( `if`(irem(j, 2)=t, b(n-1, max(m, j), 1-t), 0), j=1..m+1)) end: T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, 0, 1)): seq(T(n), n=0..12); -
Mathematica
b[n_, m_, t_] := b[n, m, t] = If[n==0, x^m, Sum[If[Mod[j, 2]==t, b[n-1, Max[m, j], 1-t], 0], {j, 1, m+1}]]; T[n_] := Function [p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, 0, 1]]; Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Dec 18 2016, after Alois P. Heinz *)
Formula
Sum_{k=0..n} k * T(n,k) = A364267(n). - Alois P. Heinz, Jul 16 2023
Comments