A368638 a(n) is the number of triangular partitions whose Young diagram fits inside a square of side n.

1, 2, 5, 12, 25, 48, 83, 136, 211, 314, 449, 626, 849, 1130, 1475, 1892, 2389, 2982, 3677, 4492, 5435, 6518, 7751, 9156, 10741, 12526, 14523, 16750, 19219, 21958, 24975, 28300, 31949, 35942, 40295, 45032, 50165, 55730, 61745, 68234, 75213, 82722, 90773, 99408

Offset: 0

Author

Alejandro B. Galván, Jan 01 2024

Keywords

• nonn

Comments

Equivalently, a(n) is the number of triangular subpartitions of the staircase partition (n, n-1, ..., 1).

Links

• Sergi Elizalde and Alejandro B. Galván, Triangular partitions: enumeration, structure, and generation, arXiv:2312.16353 [math.CO], (2023).

Crossrefs

The number of triangular partitions of size n is in A352882.

Programs

MATLAB

% subpart(n) := a(n-1).
nmax = 44;
for n = 1 : nmax
    subpart(n) = 1;
    for i = 1 : n
        subpart(n) = subpart(n) + (n - i + 1)*(n - i)*eulerPhi(i)/2;
    end
end

Formula

a(n) = 1 + Sum_{i=1..n} binomial(n-i+2,2)*phi(i).