|
|
A274581
|
|
Number T(n,k) of set partitions of [n] with alternating parity of elements and exactly k blocks; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
|
|
13
|
|
|
1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 1, 0, 1, 5, 7, 4, 1, 0, 1, 7, 14, 12, 5, 1, 0, 1, 11, 30, 33, 19, 6, 1, 0, 1, 15, 57, 84, 62, 27, 7, 1, 0, 1, 23, 119, 222, 204, 108, 37, 8, 1, 0, 1, 31, 224, 545, 627, 409, 169, 48, 9, 1, 0, 1, 47, 460, 1425, 2006, 1558, 763, 254, 61, 10, 1
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,9
|
|
LINKS
|
|
|
FORMULA
|
Sum_{k=0..n} k * T(n,k) = A305823(n).
|
|
EXAMPLE
|
T(5,1) = 1: 12345.
T(5,2) = 5: 1234|5, 123|45, 12|345, 145|23, 1|2345.
T(5,3) = 7: 123|4|5, 12|34|5, 12|3|45, 1|234|5, 145|2|3, 1|2|345, 1|23|45.
T(5,4) = 4: 12|3|4|5, 1|23|4|5, 1|2|34|5, 1|2|3|45.
T(5,5) = 1: 1|2|3|4|5.
Triangle T(n,k) begins:
1;
0, 1;
0, 1, 1;
0, 1, 2, 1;
0, 1, 3, 3, 1;
0, 1, 5, 7, 4, 1;
0, 1, 7, 14, 12, 5, 1;
0, 1, 11, 30, 33, 19, 6, 1;
0, 1, 15, 57, 84, 62, 27, 7, 1;
0, 1, 23, 119, 222, 204, 108, 37, 8, 1;
0, 1, 31, 224, 545, 627, 409, 169, 48, 9, 1;
...
|
|
MAPLE
|
b:= proc(l, i, t) option remember; `if`(l=[], x,
`if`(l[1]=t, 0, expand(x*b(subsop(1=[][], l), 1, 1-t)
))+add(`if`(l[j]=t, 0, b(subsop(j=[][], l), j, 1-t)
), j=i..nops(l)))
end:
T:= n-> `if`(n=0, 1, (p-> seq(coeff(p, x, j), j=0..n))(
b([seq(irem(i, 2), i=2..n)], 1$2))):
seq(T(n), n=0..12);
|
|
MATHEMATICA
|
b[l_, i_, t_] := b[l, i, t] = If[l == {}, x, If[l[[1]] == t, 0, Expand[x*b[Rest[l], 1, 1 - t]]] + Sum[If[l[[j]] == t, 0, b[Delete[l, j], j, 1 - t]], {j, i, Length[l]}]];
T[n_] := If[n==0, {1}, Function[p, Table[Coefficient[p, x, j], {j, 0, n}]][ b[Table[Mod[i, 2], {i, 2, n}], 1, 1]]];
|
|
CROSSREFS
|
Columns k=0-10 give: A000007, A057427, A052955(n-2) for n>1, A305777, A305778, A305779, A305780, A305781, A305782, A305783, A305784.
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|