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

0,14

Comments

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

Links

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

Example

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;

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 *)

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).

Sequence in context: A308999 A208343 A029324 * A029318 A358402 A210381

Adjacent sequences: A227548 A227549 A227550 * A227552 A227553 A227554

Keyword

nonn,look,tabf

Author

Alois P. Heinz, Jul 16 2013

Status

approved