A264033 - OEIS

%I #26 Feb 03 2017 09:51:54

%S 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 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 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 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 Row sums give A000041.

%C T(n,0) gives A000005(n) for n>0. - _Alois P. Heinz_, Nov 01 2015

%D Richard Stanley, Enumerative combinatorics. Vol. 2 MathSciNet:1676282, page 375.

%H Alois P. Heinz, <a href="/A264033/b264033.txt">Rows n = 0..80, flattened</a>

%H 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 Sum_{k>0} k * T(n,k) = A271370(n). - _Alois P. Heinz_, Apr 05 2016

%e Triangle begins:

%e 1;

%e 1;

%e 2;

%e 2,1;

%e 3,1,1;

%e 2,2,2,1;

%e 4,2,1,2,2;

%e 2,3,3,3,1,2,1;

%e 4,3,3,3,2,3,2,1,1;

%e 3,4,3,5,2,5,2,2,1,2,1;

%e 4,4,4,7,3,4,2,4,5,1,0,2,2;

%e 2,5,5,8,2,9,4,4,3,4,1,4,1,1,2,1;

%e 6,5,4,9,4,9,4,6,5,7,2,4,3,1,2,2,2,1,1;

%e ...

%p b:= proc(n, i, p, t) option remember; expand(

%p       `if`(n=0, x^t, `if`(i<1, 0, add(

%p        b(n-i*j, i-1, p+j, t+j*p), j=0..n/i))))

%p     end:

%p T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2, 0$2)):

%p seq(T(n), n=0..15);  # _Alois P. Heinz_, Nov 01 2015

%t 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 Cf. A000005, A000041, A130519, A271370.

%K nonn,tabf

%O 0,3

%A _Christian Stump_, Nov 01 2015