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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.
11
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
0,19
COMMENTS
All odd elements are in blocks with an odd index and all even elements are in blocks with an even index.
LINKS
FORMULA
Sum_{k=0..n} k * T(n,k) = A364267(n). - Alois P. Heinz, Jul 16 2023
EXAMPLE
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;
...
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 *)
CROSSREFS
Row sums give A274538.
Columns k=0-10 give: A000007, A000007(n-1), A000012(n-2), A052551(n-3), A274868, A274869, A274870, A274871, A274872, A274873, A274874.
T(2n,n) gives A274875.
Main diagonal and lower diagonals give: A000012, A004526, A002623(n-2) or A173196.
Cf. A364267.
Sequence in context: A330959 A083199 A327187 * A305234 A021761 A221960
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Jun 27 2016
STATUS
approved