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
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
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 *)
Comments