A259478 Partition containment triangle.

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

Alois P. Heinz, Rows n = 1..141, flattened

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 *)

Crossrefs

Columns k=1-10 give: A000041, 2*A000065, A303853, A303854, A303855, A303856, A303857, A303858, A303859, A303860.

Cf. A000070, A259479, A259480, A259481, A161492, A227309, A006958, A297388, A303851, A303852, A303861, A303862, A303863.

Sequence in context: A096858 A037254 A316939 * A155706 A317533 A360999

Adjacent sequences: A259475 A259476 A259477 * A259479 A259480 A259481

Keyword

nonn,tabl

Author

Wouter Meeussen, Jun 28 2015

Status

approved