A270768 -id:A270768

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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.

+10
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

Alois P. Heinz, Rows n = 1..141, flattened

Wikipedia, Partition of a set

FORMULA

T(n,k) = A270702(n,n-k+1).

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

Columns k=1-10 give: A000110(n-1), A270756, A270757, A270758, A270759, A270760, A270761, A270762, A270763, A270764.

Main and lower diagonals give: A124427, A270765, A270766, A270767, A270768, A270769, A270770, A270771, A270772, A270773.

Row sums give A070071.

Reflected triangle gives A270702.

T(2n-1,n) gives A270703.

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Mar 21 2016

STATUS

approved

A270702

Total sum T(n,k) of the sizes of all blocks with minimal element k in all set partitions of {1,2,...,n}; triangle T(n,k), n>=1, 1<=k<=n, read by rows.

+10
23

1, 3, 1, 9, 4, 2, 30, 16, 9, 5, 112, 67, 41, 25, 15, 463, 299, 195, 127, 82, 52, 2095, 1429, 979, 670, 456, 307, 203, 10279, 7307, 5204, 3702, 2623, 1845, 1283, 877, 54267, 39848, 29278, 21485, 15717, 11437, 8257, 5894, 4140, 306298, 230884, 174029, 131007, 98367, 73561, 54692, 40338, 29427, 21147

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Alois P. Heinz, Rows n = 1..141, flattened

Wikipedia, Partition of a set

FORMULA

T(n,k) = A270701(n,n-k+1).

EXAMPLE

Row n=3 is [9, 4, 2] = [3+2+2+1+1, 0+0+1+2+1, 0+1+0+0+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;

: 3, 1;

: 9, 4, 2;

: 30, 16, 9, 5;

: 112, 67, 41, 25, 15;

: 463, 299, 195, 127, 82, 52;

: 2095, 1429, 979, 670, 456, 307, 203;

: 10279, 7307, 5204, 3702, 2623, 1845, 1283, 877;

: 54267, 39848, 29278, 21485, 15717, 11437, 8257, 5894, 4140;