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A264033 Triangle read by rows: T(n,k) (n>=0, 0<=k<=A130519(n+1)) is the number of integer partitions of n having k pairs of different size.

%I A264033 #26 Feb 03 2017 09:51:54
%S A264033 1,1,2,2,1,3,1,1,2,2,2,1,4,2,1,2,2,2,3,3,3,1,2,1,4,3,3,3,2,3,2,1,1,3,
%T A264033 4,3,5,2,5,2,2,1,2,1,4,4,4,7,3,4,2,4,5,1,0,2,2,2,5,5,8,2,9,4,4,3,4,1,
%U A264033 4,1,1,2,1,6,5,4,9,4,9,4,6,5,7,2,4,3,1,2,2,2,1,1
%N A264033 Triangle read by rows: T(n,k) (n>=0, 0<=k<=A130519(n+1)) is the number of integer partitions of n having k pairs of different size.
%C A264033 Row sums give A000041.
%C A264033 T(n,0) gives A000005(n) for n>0. - _Alois P. Heinz_, Nov 01 2015
%D A264033 Richard Stanley, Enumerative combinatorics. Vol. 2 MathSciNet:1676282, page 375.
%H A264033 Alois P. Heinz, <a href="/A264033/b264033.txt">Rows n = 0..80, flattened</a>
%H A264033 FindStat - Combinatorial Statistic Finder, <a href="http://www.findstat.org/StatisticsDatabase/St000175">Degree of the polynomial counting the number of semistandard Young-tableaux of shape k*lambda</a>.
%F A264033 Sum_{k>0} k * T(n,k) = A271370(n). - _Alois P. Heinz_, Apr 05 2016
%e A264033 Triangle begins:
%e A264033 1;
%e A264033 1;
%e A264033 2;
%e A264033 2,1;
%e A264033 3,1,1;
%e A264033 2,2,2,1;
%e A264033 4,2,1,2,2;
%e A264033 2,3,3,3,1,2,1;
%e A264033 4,3,3,3,2,3,2,1,1;
%e A264033 3,4,3,5,2,5,2,2,1,2,1;
%e A264033 4,4,4,7,3,4,2,4,5,1,0,2,2;
%e A264033 2,5,5,8,2,9,4,4,3,4,1,4,1,1,2,1;
%e A264033 6,5,4,9,4,9,4,6,5,7,2,4,3,1,2,2,2,1,1;
%e A264033 ...
%p A264033 b:= proc(n, i, p, t) option remember; expand(
%p A264033       `if`(n=0, x^t, `if`(i<1, 0, add(
%p A264033        b(n-i*j, i-1, p+j, t+j*p), j=0..n/i))))
%p A264033     end:
%p A264033 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2, 0$2)):
%p A264033 seq(T(n), n=0..15);  # _Alois P. Heinz_, Nov 01 2015
%t A264033 b[n_, i_, p_, t_] := b[n, i, p, t] = Expand[If[n==0, x^t, If[i<1, 0, Sum[b[n-i*j, i-1, p+j, t+j*p], {j, 0, n/i}]]]]; T[n_] := Function [p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, n, 0, 0]]; Table[T[n], {n, 0, 15}] // Flatten (* _Jean-François Alcover_, Feb 03 2017, after _Alois P. Heinz_ *)
%Y A264033 Cf. A000005, A000041, A130519, A271370.
%K A264033 nonn,tabf
%O A264033 0,3
%A A264033 _Christian Stump_, Nov 01 2015