A262495 Number T(n,k) of partitions of n into parts of exactly k sorts which are introduced in ascending order such that sorts of adjacent parts are different; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 4, 4, 1, 0, 1, 6, 12, 7, 1, 0, 1, 10, 31, 33, 11, 1, 0, 1, 14, 73, 130, 77, 16, 1, 0, 1, 21, 165, 464, 438, 157, 22, 1, 0, 1, 29, 357, 1558, 2216, 1223, 289, 29, 1, 0, 1, 41, 760, 5039, 10423, 8331, 2957, 492, 37, 1

Offset: 0

Author

Alois P. Heinz, Sep 24 2015

Keywords

• nonn
• tabl

Examples

			T(3,1) = 1: 3a.
T(3,2) = 2: 2a1b, 1a1b1a.
T(3,3) = 1: 1a1b1c.
T(5,3) = 12: 3a1b1c, 2a2b1c, 2a1b1a1c, 2a1b1c1a, 2a1b1c1b, 1a1b1a1b1c, 1a1b1a1c1a, 1a1b1a1c1b, 1a1b1c1a1b, 1a1b1c1a1c, 1a1b1c1b1a, 1a1b1c1b1c.
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1,  1;
  0, 1,  2,   1;
  0, 1,  4,   4,    1;
  0, 1,  6,  12,    7,     1;
  0, 1, 10,  31,   33,    11,    1;
  0, 1, 14,  73,  130,    77,   16,    1;
  0, 1, 21, 165,  464,   438,  157,   22,   1;
  0, 1, 29, 357, 1558,  2216, 1223,  289,  29,  1;
  0, 1, 41, 760, 5039, 10423, 8331, 2957, 492, 37, 1;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..140, flattened

Crossrefs

Columns k=0-10 give: A000007, A057427, A000065, A262445/6 = A320545, A320546, A320547, A320548, A320549, A320550, A320551, A320552.

Main diagonal and lower diagonal give: A000012, A000124 (shifted).

Row sums give A262496.

T(2n,n) gives A262529.

Cf. A256130.

Programs

Maple

b:= proc(n, i, k) option remember; `if`(n=0 or i=1, k^(n-1),
      b(n, i-1, k) +`if`(i>n, 0, k*b(n-i, i, k)))
    end:
A:= (n, k)-> `if`(n=0, 1, `if`(k<2, k, k*b(n$2, k-1))):
T:= (n, k)-> add(A(n, k-i)*(-1)^i/(i!*(k-i)!), i=0..k):
seq(seq(T(n, k), k=0..n), n=0..12);

Mathematica

b[n_, i_, k_] := b[n, i, k] = If[n == 0 || i == 1, k^(n-1), b[n, i-1, k] + If[i>n, 0, k*b[n-i, i, k]]]; A[n_, k_] := If[n == 0, 1, If[k<2, k, k*b[n, n, k-1]]]; T[n_, k_] := Sum[A[n, k-i]*(-1)^i/(i!*(k-i)!), {i, 0, k}]; Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 24 2017, translated from Maple *)