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
Alois P. Heinz, Rows n = 1..141, flattened
Eric Weisstein's World of Mathematics, Ferrers Diagram
Wikipedia, Ferrers diagram
FORMULA
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 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Wouter Meeussen, Jun 28 2015
STATUS
approved