A225751 Number of different figures obtained by a putting two Young diagrams of partitions lambda and mu, such that |lambda| + |mu| = n on top of each other.

1, 1, 3, 5, 10, 15, 26, 38, 60, 85, 127, 176, 253, 343, 478, 639, 870, 1145, 1530, 1990, 2617, 3367, 4369, 5568, 7143, 9024, 11460, 14369, 18087, 22517, 28121, 34787, 43136, 53048, 65358, 79944, 97921, 119173, 145188, 175883, 213221, 257177, 310351, 372820

Offset

0,3

Comments

See Mukhin reference item 4.3. 'A challenge', sum of p(k) for k = ceiling(n/2) to n with p(k) the partition numbers A000041. Remark that the indexing in the reference misses N_3 (should be N_3=5 and so on).

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Eugene Mukhin, Symmetric polynomials and partitions

Formula

a(n) = Sum_{i=ceiling(n/2)..n} A000041(i).

a(n) = A000070(n) - A000070(ceiling(n/2)-1). - Alois P. Heinz, Jul 31 2016

Example

a(3) = 5 is illustrated by the following 5 different results:
{2}   = {1}  &  {2}
{2}   = {2}  &  {1}
{3}   = { }  &  {3}
{1,1} = {1}  &  {1,1}
{1,1} = {1,1}&  {1}
{2,1} =  { } &  {2,1}
{1,1,1}= { } &  {1,1,1}
producing {2}, {3}, {1,1}, {2,1} and {1,1,1} as superpositions of two partitions with sum of lengths = 3.

Maple

with(combinat):
a:= n-> add(numbpart(i), i=ceil(n/2)..n):
seq(a(n), n=0..50);  # Alois P. Heinz, May 15 2013

Mathematica

Table[Sum[PartitionsP[k], {k, Ceiling[n/2], n}], {n, 36}]

Prog

(PARI) a(n)=sum(k=ceil(n/2), n, numbpart(k)); \\ Joerg Arndt, May 15 2013

Crossrefs

Cf. A000041, A000070.

Sequence in context: A090491 A126728 A070557 * A264397 A254346 A132302

Adjacent sequences: A225748 A225749 A225750 * A225752 A225753 A225754

Keyword

nonn

Author

Wouter Meeussen, May 14 2013

Status

approved