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%I #47 May 02 2018 10:53:30
%S 1,2,2,3,4,3,5,8,7,5,7,12,13,12,7,11,20,23,25,19,11,15,28,35,42,39,30,
%T 15,22,42,54,70,70,66,45,22,30,58,78,105,114,119,99,67,30,42,82,112,
%U 158,178,202,186,155,97,42,56,110,154,223,262,313,314,292,226,139,56,77,152,215,319,383,479,503,511,442,336,195,77
%N Partition containment triangle.
%C T(n,k) counts pairs of partitions (lambda,mu) with Ferrers diagram of mu not extending beyond the diagram of lambda for all partitions lambda of size n and mu of size k <= n.
%C First column and main diagonal both equal A000041 (partition numbers).
%C This sequence counts (2,1)/(1) as different from (3,2,1)/(3,1) while their set-theoretic difference lambda - mu (their skew diagram) is the same.
%D I. G. MacDonald: "Symmetric functions and Hall polynomials", Oxford University Press, 1979. Page 4.
%H Alois P. Heinz, <a href="/A259478/b259478.txt">Rows n = 1..141, flattened</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FerrersDiagram.html">Ferrers Diagram</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Ferrers_diagram">Ferrers diagram</a>
%F Sum_{k=1..n} T(n,k) = A297388(n) - A000041(n). - _Alois P. Heinz_, Jan 10 2018
%e T(3,2) = 4, the pairs of partitions are ((3)/(2)), ((2,1)/(2), ((2,1)/(1,1)), ((1,1,1)/(1,1))
%e and the diagrams are:
%e x x 0 , x x , x 0 , x
%e 0 x x
%e 0
%e triangle begins:
%e n=1; 1
%e n=2; 2 2
%e n=3; 3 4 3
%e n=4; 5 8 7 5
%e n=5; 7 12 13 12 7
%e n=6; 11 20 23 25 19 11
%p b:= proc(n, i, t) option remember; expand(`if`(n=0 or i=1,
%p `if`(t=0, 1, add(x^j, j=0..n)), b(n, i-1, min(i-1, t))+
%p add(b(n-i, min(i, n-i), min(j, n-i))*x^j, j=0..t)))
%p end:
%p T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n$3)):
%p seq(T(n), n=1..15); # _Alois P. Heinz_, Jul 05 2015, revised Jan 10 2018
%t majorsweak[left_List,right_List]:=Block[{le1=Length[left],le2=Length[right]},If[le2>le1||Min[Sign[left-PadRight[right,le1]]]<0,False,True]];
%t Table[Sum[ If[! majorsweak[\[Lambda], \[Mu]], 0, 1] , {\[Lambda], IntegerPartitions[n] }, {\[Mu], IntegerPartitions[m]}], {n, 7}, {m, n}]
%t (* Second program: *)
%t b[n_, m_, i_, j_, t_] := b[n, m, i, j, t] = If[m > n, 0, If[n == 0, 1, If[i < 1, 0, If[t && j > 0, b[n, m, i, j - 1, t], 0] + If[i > j, b[n, m, i - 1, j, False], 0] + If[i > n || j > m, 0, b[n - i, m - j, i, j, True]]]]]; T[n_, m_] := b[n, m, n, m, True]; Table[Table[T[n, m], {m, 1, n}], {n, 1, 14}] // Flatten (* _Jean-François Alcover_, Aug 27 2016, after _Alois P. Heinz_ *)
%Y Columns k=1-10 give: A000041, 2*A000065, A303853, A303854, A303855, A303856, A303857, A303858, A303859, A303860.
%Y Cf. A000070, A259479, A259480, A259481, A161492, A227309, A006958, A297388, A303851, A303852, A303861, A303862, A303863.
%K nonn,tabl
%O 1,2
%A _Wouter Meeussen_, Jun 28 2015
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