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A363493
Number T(n,k) of partitions of [n] having exactly k parity changes within their blocks, n>=0, 0<=k<=max(0,n-1), read by rows.
7
1, 1, 1, 1, 2, 2, 1, 4, 6, 4, 1, 10, 18, 17, 6, 1, 25, 61, 68, 38, 10, 1, 75, 210, 292, 202, 83, 14, 1, 225, 778, 1252, 1116, 576, 170, 22, 1, 780, 3008, 5670, 5928, 3899, 1490, 341, 30, 1, 2704, 12219, 26114, 32382, 25320, 12655, 3856, 678, 46, 1, 10556, 52268, 126073, 177666, 163695, 98282, 39230, 9418, 1319, 62, 1
OFFSET
0,5
LINKS
FORMULA
Sum_{k=0..max(0,n-1)} k * T(n,k) = A363496(n).
EXAMPLE
T(4,0) = 4: 13|24, 13|2|4, 1|24|3, 1|2|3|4.
T(4,1) = 6: 124|3, 12|3|4, 134|2, 1|23|4, 14|2|3, 1|2|34.
T(4,2) = 4: 123|4, 12|34, 14|23, 1|234.
T(4,3) = 1: 1234.
T(5,2) = 17: 1235|4, 123|4|5, 1245|3, 12|34|5, 125|3|4, 12|3|45, 1345|2, 134|25, 14|235, 14|23|5, 15|234, 1|234|5, 1|23|45, 145|2|3, 14|25|3, 1|25|34, 1|2|345.
Triangle T(n,k) begins:
1;
1;
1, 1;
2, 2, 1;
4, 6, 4, 1;
10, 18, 17, 6, 1;
25, 61, 68, 38, 10, 1;
75, 210, 292, 202, 83, 14, 1;
225, 778, 1252, 1116, 576, 170, 22, 1;
780, 3008, 5670, 5928, 3899, 1490, 341, 30, 1;
2704, 12219, 26114, 32382, 25320, 12655, 3856, 678, 46, 1;
...
MAPLE
b:= proc(n, x, y) option remember; `if`(n=0, 1,
`if`(y=0, 0, expand(b(n-1, y-1, x+1)*y*z))+
b(n-1, y, x)*x + b(n-1, y, x+1))
end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(n, 0$2)):
seq(T(n), n=0..12);
MATHEMATICA
b[n_, x_, y_] := b[n, x, y] = If[n == 0, 1,
If[y == 0, 0, Expand[b[n - 1, y - 1, x + 1]*y*z]] +
b[n - 1, y, x]*x + b[n - 1, y, x + 1]];
T[n_] := CoefficientList[b[n, 0, 0], z];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Sep 05 2023, after Alois P. Heinz *)
CROSSREFS
Columns k=0-2 give: A124419, A363511, A363588.
Row sums give A000110.
T(n+1,n) gives A000012.
T(n+2,n) gives A027383.
T(2n+1,n) gives A363495.
Sequence in context: A228336 A111062 A369632 * A193597 A191490 A061598
KEYWORD
nonn,tabf
AUTHOR
Alois P. Heinz, Jun 05 2023
STATUS
approved