A224878 Number T(n,k) of partitions of n into distinct parts with boundary size k (where one part of size 0 is allowed).

1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 3, 0, 1, 2, 1, 0, 1, 3, 2, 0, 1, 5, 2, 0, 1, 4, 5, 0, 1, 4, 6, 1, 0, 1, 6, 8, 1, 0, 1, 7, 9, 3, 0, 1, 6, 13, 4, 0, 1, 7, 15, 7, 0, 1, 7, 18, 10, 0, 1, 8, 20, 14, 1, 0, 1, 11, 23, 17, 2, 0, 1, 8, 28, 24, 3, 0, 1, 9, 31, 30, 5, 0, 1

Offset: 0

Author

Patrick Devlin, Jul 23 2013

Keywords

• nonn
• tabf

Comments

Boundary size of a partition (or set) is the number of parts (elements) having fewer than 2 neighbors.

T(n,k) is also the number of subsets of {0, 1, 2, ...} whose elements sum to n and that have k elements in its boundary.

Examples

			T(9,1) = 1: [9].
T(9,2) = 6: [0,9], [1,8], [2,7], [3,6], [4,5], [2,3,4].
T(9,3) = 8: [1,2,6], [1,3,5], [0,1,8], [0,2,7], [0,3,6], [0,4,5], [0,2,3,4], [0,1,2,6].
T(9,4) = 1: [0,1,3,5].
Triangle T(n,k) begins:
1, 1; (namely, the empty set and the set {0})
0, 1, 1;
0, 1, 1;
0, 1, 3;
0, 1, 2,  1;
0, 1, 3,  2;
0, 1, 5,  2;
0, 1, 4,  5;
0, 1, 4,  6, 1;
0, 1, 6,  8, 1;
0, 1, 7,  9, 3;
0, 1, 6, 13, 4;
0, 1, 7, 15, 7;
		

Links

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

Crossrefs

Cf. A227551 (no parts of size 0 are allowed).

Row sums are twice A000009.

Programs

Maple

b:= proc(n, i, t) option remember; `if`(n=0 and i<0, `if`(t>1, x, 1),
      expand(`if`(i<0, 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);  # Alois P. Heinz, Jul 23 2013

Mathematica

b[n_, i_, t_] := b[n, i, t] = If[n==0 && i<0, If[t>1, x, 1], Expand[If[i<0, 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, Feb 07 2017, after Alois P. Heinz *)