A276727 - OEIS

A276727

Number T(n,k) of set partitions of [n] where k is minimal such that for each block b the smallest integer interval containing b has at most k elements; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 4, 5, 5, 0, 1, 7, 12, 17, 15, 0, 1, 12, 29, 45, 64, 52, 0, 1, 20, 66, 121, 201, 265, 203, 0, 1, 33, 145, 336, 585, 966, 1197, 877, 0, 1, 54, 315, 901, 1741, 3172, 4971, 5852, 4140, 0, 1, 88, 676, 2347, 5375, 10100, 18223, 27267, 30751, 21147

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OFFSET

0,9

LINKS

Alois P. Heinz, Rows n = 0..20, flattened

FORMULA

T(n,k) = A276719(n,k) - A276719(n,k-1) for k>0, T(n,0) = A000007(n).

EXAMPLE

T(4,1) = 1: 1|2|3|4.

T(4,2) = 4: 12|34, 12|3|4, 1|23|4, 1|2|34.

T(4,3) = 5: 123|4, 13|24, 13|2|4, 1|234, 1|24|3.

T(4,4) = 5: 1234, 124|3, 134|2, 14|23, 14|2|3.

T(5,4) = 17: 1234|5, 124|35, 124|3|5, 134|25, 134|2|5, 13|245, 13|25|4, 14|235, 14|23|5, 1|2345, 1|235|4, 14|25|3, 14|2|35, 14|2|3|5, 1|245|3, 1|25|34, 1|25|3|4.

Triangle T(n,k) begins:

0, 1;

0, 1, 1;

0, 1, 2, 2;

0, 1, 4, 5, 5;

0, 1, 7, 12, 17, 15;

0, 1, 12, 29, 45, 64, 52;

0, 1, 20, 66, 121, 201, 265, 203;

0, 1, 33, 145, 336, 585, 966, 1197, 877;

...

MAPLE

b:= proc(n, m, l) option remember; `if`(n=0, 1,

add(b(n-1, max(m, j), [subsop(1=NULL, l)[],

`if`(j<=m, 0, j)]), j={l[], m+1} minus {0}))

end:

A:= (n, k)-> `if`(n=0, 1, `if`(k<2, k, b(n, 0, [0$(k-1)]))):

T:= (n, k)-> A(n, k) -`if`(k=0, 0, A(n, k-1)):

seq(seq(T(n, k), k=0..n), n=0..12);

MATHEMATICA

b[n_, m_, l_List] := b[n, m, l] = If[n == 0, 1, Sum[b[n - 1, Max[m, j], Append[ReplacePart[l, 1 -> Nothing], If[j <= m, 0, j]]], {j, Append[l, m + 1] ~Complement~ {0}}]]; A[n_, k_] := If[n == 0, 1, If[k < 2, k, b[n, 0, Array[0&, k - 1]]]]; T [n_, k_] := A[n, k] - If[k == 0, 0, A[n, k - 1]]; Table[T[n, k], {n, 0, 12}, { k, 0, n}] // Flatten (* Jean-François Alcover, Feb 04 2017, translated from Maple *)

CROSSREFS

Columns k=0-10 give: A000007, A057427, A000071(n+1), A320553, A320554, A320555, A320556, A320557, A320558, A320559, A320560.

Row sums give A000110.

Main diagonal gives A000110(n-1) for n>0.

T(2n,n) gives A276728.

Cf. A263757, A276719, A276891.

Sequence in context: A317575 A295653 A146326 * A267617 A158852 A188285

Adjacent sequences: A276724 A276725 A276726 * A276728 A276729 A276730

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Sep 16 2016

STATUS

approved