A276891 Number T(n,k) of ordered 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, 2, 1, 0, 6, 4, 3, 0, 24, 20, 18, 13, 0, 120, 114, 118, 114, 75, 0, 720, 750, 878, 924, 870, 541, 0, 5040, 5616, 7224, 8152, 8760, 7818, 4683, 0, 40320, 47304, 65514, 79682, 90084, 94560, 81078, 47293, 0, 362880, 443400, 652446, 845874, 998560, 1135776, 1148016, 954474, 545835

Offset: 0

Author

Alois P. Heinz, Sep 21 2016

Keywords

• nonn
• tabl

Examples

			Triangle T(n,k) begins:
  1;
  0,     1;
  0,     2,     1;
  0,     6,     4,     3;
  0,    24,    20,    18,    13;
  0,   120,   114,   118,   114,    75;
  0,   720,   750,   878,   924,   870,   541;
  0,  5040,  5616,  7224,  8152,  8760,  7818,  4683;
  0, 40320, 47304, 65514, 79682, 90084, 94560, 81078, 47293;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..21, flattened

Crossrefs

Columns k=0-10 give: A000007, A000142 (for n>0), A320615, A320616, A320617, A320618, A320619, A320620, A320621, A320622, A320623.

Row sums give: A000670.

Main diagonal gives A000670(n-1) for n>0.

T(2n,n) gives A276892.

Cf. A276727, A276890.

Programs

Maple

b:= proc(n, m, l) option remember; `if`(n=0, m!,
      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`(k=0, `if`(n=0, 1, 0),
             `if`(k=1, n!, 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..10);

Mathematica

b[n_, m_, l_List] := b[n, m, l] = If[n == 0, m!, 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[k == 0, If[n == 0, 1, 0], If[k == 1, n!, 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, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 04 2017, translated from Maple *)

Formula

T(n,k) = A276890(n,k) - A276890(n,k-1) for k>0, T(n,0) = A000007(n).