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

0,2

Comments

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

Links

Table of n, a(n) for n=0..43.

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

Formula

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

Prog

(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

Crossrefs

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

Sequence in context: A116720 A116731 A116722 * A116730 A240847 A166106

Adjacent sequences: A368635 A368636 A368637 * A368639 A368640 A368641

Keyword

nonn

Author

Alejandro B. Galván, Jan 01 2024

Status

approved