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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.
13
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
OFFSET
0,9
LINKS
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:
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;
...
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
Row sums give A000110.
Main diagonal gives A000110(n-1) for n>0.
T(2n,n) gives A276728.
Sequence in context: A317575 A295653 A146326 * A267617 A158852 A188285
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Sep 16 2016
STATUS
approved