A227551 Number T(n,k) of partitions of n into distinct parts with boundary size k; triangle T(n,k), n>=0, 0<=k<=A227568(n), read by rows.

1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 2, 0, 1, 3, 0, 1, 3, 1, 0, 1, 3, 2, 0, 1, 5, 2, 0, 1, 5, 4, 0, 1, 5, 6, 0, 1, 6, 7, 1, 0, 1, 6, 10, 1, 0, 1, 7, 11, 3, 0, 1, 9, 13, 4, 0, 1, 7, 18, 6, 0, 1, 8, 20, 9, 0, 1, 10, 21, 14, 0, 1, 9, 27, 16, 1, 0, 1, 10, 29, 22, 2

Offset: 0

Author

Alois P. Heinz, Jul 16 2013

Keywords

• nonn
• look
• tabf

Comments

The boundary size is the number of parts having fewer than two neighbors.

Examples

			T(12,1) = 1: [12].
T(12,2) = 6: [1,11], [2,10], [3,4,5], [3,9], [4,8], [5,7].
T(12,3) = 7: [1,2,3,6], [1,2,9], [1,3,8], [1,4,7], [1,5,6], [2,3,7], [2,4,6].
T(12,4) = 1: [1,2,4,5].
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1;
  0, 1, 1;
  0, 1, 1;
  0, 1, 2;
  0, 1, 3;
  0, 1, 3, 1;
  0, 1, 3, 2;
  0, 1, 5, 2;
  0, 1, 5, 4;
  0, 1, 5, 6;
  0, 1, 6, 7, 1;
		

Links

• Alois P. Heinz, Rows n = 0..600, flattened

Crossrefs

Columns k=0-10 give: A000007, A057427, A227559, A227560, A227561, A227562, A227563, A227564, A227565, A227566, A227567.

Row sums give: A000009.

Last elements of rows give: A227552.

Cf. A227345 (a version with trailing zeros), A053993, A201077, A227568, A224878 (one part of size 0 allowed).

Programs

Maple

b:= proc(n, i, t) option remember; `if`(n=0, `if`(t>1, x, 1),
      expand(`if`(i<1, 0, `if`(t>1, x, 1)*b(n, i-1, iquo(t, 2))+
      `if`(i>n, 0, `if`(t=2, x, 1)*b(n-i, i-1, iquo(t, 2)+2)))))
    end:
T:= n-> (p->seq(coeff(p, x, i), i=0..degree(p)))(b(n$2, 0)):
seq(T(n), n=0..30);

Mathematica

b[n_, i_, t_] := b[n, i, t] = If[n == 0, If[t > 1, x, 1], Expand[If[i < 1, 0, If[t > 1, x, 1]*b[n, i - 1, Quotient[t, 2]] + If[i > n, 0, If[t == 2, x, 1]*b[n - i, i - 1, Quotient[t, 2] + 2]]]]]; T[n_] := Function [p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, n, 0]]; Table[T[n], {n, 0, 30}] // Flatten (* Jean-François Alcover, Dec 12 2016, after Alois P. Heinz *)