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

%I A224878 #15 Feb 07 2017 06:40:36
%S A224878 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,
%T A224878 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,
%U A224878 11,23,17,2,0,1,8,28,24,3,0,1,9,31,30,5,0,1
%N A224878 Number T(n,k) of partitions of n into distinct parts with boundary size k (where one part of size 0 is allowed).
%C A224878 Boundary size of a partition (or set) is the number of parts (elements) having fewer than 2 neighbors.
%C A224878 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.
%H A224878 Alois P. Heinz, <a href="/A224878/b224878.txt">Rows n = 0..600, flattened</a>
%e A224878 T(9,1) = 1: [9].
%e A224878 T(9,2) = 6: [0,9], [1,8], [2,7], [3,6], [4,5], [2,3,4].
%e A224878 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].
%e A224878 T(9,4) = 1: [0,1,3,5].
%e A224878 Triangle T(n,k) begins:
%e A224878 1, 1; (namely, the empty set and the set {0})
%e A224878 0, 1, 1;
%e A224878 0, 1, 1;
%e A224878 0, 1, 3;
%e A224878 0, 1, 2,  1;
%e A224878 0, 1, 3,  2;
%e A224878 0, 1, 5,  2;
%e A224878 0, 1, 4,  5;
%e A224878 0, 1, 4,  6, 1;
%e A224878 0, 1, 6,  8, 1;
%e A224878 0, 1, 7,  9, 3;
%e A224878 0, 1, 6, 13, 4;
%e A224878 0, 1, 7, 15, 7;
%p A224878 b:= proc(n, i, t) option remember; `if`(n=0 and i<0, `if`(t>1, x, 1),
%p A224878       expand(`if`(i<0, 0, `if`(t>1, x, 1)*b(n, i-1, iquo(t, 2))+
%p A224878       `if`(i>n, 0, `if`(t=2, x, 1)*b(n-i, i-1, iquo(t, 2)+2)))))
%p A224878     end:
%p A224878 T:= n-> (p->seq(coeff(p, x, i), i=0..degree(p)))(b(n$2, 0)):
%p A224878 seq(T(n), n=0..30);  # _Alois P. Heinz_, Jul 23 2013
%t A224878 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_ *)
%Y A224878 Cf. A227551 (no parts of size 0 are allowed).
%Y A224878 Row sums are twice A000009.
%K A224878 nonn,tabf
%O A224878 0,11
%A A224878 _Patrick Devlin_, Jul 23 2013