|
|
A270701
|
|
Total sum T(n,k) of the sizes of all blocks with maximal element k in all set partitions of {1,2,...,n}; triangle T(n,k), n>=1, 1<=k<=n, read by rows.
|
|
23
|
|
|
1, 1, 3, 2, 4, 9, 5, 9, 16, 30, 15, 25, 41, 67, 112, 52, 82, 127, 195, 299, 463, 203, 307, 456, 670, 979, 1429, 2095, 877, 1283, 1845, 2623, 3702, 5204, 7307, 10279, 4140, 5894, 8257, 11437, 15717, 21485, 29278, 39848, 54267, 21147, 29427, 40338, 54692, 73561, 98367, 131007, 174029, 230884, 306298
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,3
|
|
LINKS
|
|
|
FORMULA
|
|
|
EXAMPLE
|
Row n=3 is [2, 4, 9] = [0+0+0+1+1, 0+2+1+0+1, 3+1+2+2+1] because the set partitions of {1,2,3} are: 123, 12|3, 13|2, 1|23, 1|2|3.
Triangle T(n,k) begins:
: 1;
: 1, 3;
: 2, 4, 9;
: 5, 9, 16, 30;
: 15, 25, 41, 67, 112;
: 52, 82, 127, 195, 299, 463;
: 203, 307, 456, 670, 979, 1429, 2095;
: 877, 1283, 1845, 2623, 3702, 5204, 7307, 10279;
: 4140, 5894, 8257, 11437, 15717, 21485, 29278, 39848, 54267;
|
|
MAPLE
|
b:= proc(n, m, t) option remember; `if`(n=0, [1, 0], add(
`if`(t=1 and j<>m+1, 0, (p->p+`if`(j=-t or t=1 and j=m+1,
[0, p[1]], 0))(b(n-1, max(m, j), `if`(t=1 and j=m+1, -j,
`if`(t<0, t, `if`(t>0, t-1, 0)))))), j=1..m+1))
end:
T:= (n, k)-> b(n, 0, max(0, 1+n-k))[2]:
seq(seq(T(n, k), k=1..n), n=1..12);
|
|
MATHEMATICA
|
b[n_, m_, t_] := b[n, m, t] = If[n == 0, {1, 0}, Sum[If[t == 1 && j != m+1, 0, Function[p, p + If[j == -t || t == 1 && j == m+1, {0, p[[1]]}, 0]][b[ n-1, Max[m, j], If[t == 1 && j == m+1, -j, If[t < 0, t, If[t > 0, t-1, 0] ]]]]], {j, 1, m+1}]];
T[n_, k_] := b[n, 0, Max[0, 1+n-k]][[2]];
Table[Table[T[n, k], {k, 1, n}], {n, 1, 12}] // Flatten (* Jean-François Alcover, Apr 24 2016, translated from Maple *)
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|