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A259478
Partition containment triangle.
19
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, 15, 22, 42, 54, 70, 70, 66, 45, 22, 30, 58, 78, 105, 114, 119, 99, 67, 30, 42, 82, 112, 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
OFFSET
1,2
COMMENTS
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.
First column and main diagonal both equal A000041 (partition numbers).
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.
REFERENCES
I. G. MacDonald: "Symmetric functions and Hall polynomials", Oxford University Press, 1979. Page 4.
LINKS
Eric Weisstein's World of Mathematics, Ferrers Diagram
Wikipedia, Ferrers diagram
FORMULA
Sum_{k=1..n} T(n,k) = A297388(n) - A000041(n). - Alois P. Heinz, Jan 10 2018
EXAMPLE
T(3,2) = 4, the pairs of partitions are ((3)/(2)), ((2,1)/(2), ((2,1)/(1,1)), ((1,1,1)/(1,1))
and the diagrams are:
x x 0 , x x , x 0 , x
0 x x
0
triangle begins:
n=1; 1
n=2; 2 2
n=3; 3 4 3
n=4; 5 8 7 5
n=5; 7 12 13 12 7
n=6; 11 20 23 25 19 11
MAPLE
b:= proc(n, i, t) option remember; expand(`if`(n=0 or i=1,
`if`(t=0, 1, add(x^j, j=0..n)), b(n, i-1, min(i-1, t))+
add(b(n-i, min(i, n-i), min(j, n-i))*x^j, j=0..t)))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n$3)):
seq(T(n), n=1..15); # Alois P. Heinz, Jul 05 2015, revised Jan 10 2018
MATHEMATICA
majorsweak[left_List, right_List]:=Block[{le1=Length[left], le2=Length[right]}, If[le2>le1||Min[Sign[left-PadRight[right, le1]]]<0, False, True]];
Table[Sum[ If[! majorsweak[\[Lambda], \[Mu]], 0, 1] , {\[Lambda], IntegerPartitions[n] }, {\[Mu], IntegerPartitions[m]}], {n, 7}, {m, n}]
(* Second program: *)
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 *)
KEYWORD
nonn,tabl
AUTHOR
Wouter Meeussen, Jun 28 2015
STATUS
approved