A352882 -id:A352882 - cp's OEIS frontend

A368554 a(n) is the number of triangular partitions of size n with two removable cells.

0, 0, 1, 2, 2, 5, 6, 4, 8, 11, 8, 14, 14, 10, 17, 22, 16, 20, 22, 20, 29, 32, 20, 32, 34, 28, 38, 39, 30, 50, 48, 32, 42, 48, 40, 61, 62, 42, 50, 66, 50, 68, 62, 54, 77, 78, 54, 74, 78, 64, 84, 92, 56, 86, 93, 84, 102, 98, 66, 100, 104, 80, 100, 110, 96, 129

Offset: 1

Alejandro B. Galván, Dec 29 2023

nonn

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

The number of triangular partitions of size n with any number of removable cells is in A352882.

Cf. A368556 (with just one removable cell).

See the g.f. in Proposition 6.1. of Elizalde and Galván.

A368556 a(n) is the number of triangular partitions of size n with just one removable cell.

Original entry on oeis.org

1, 2, 2, 2, 4, 2, 2, 6, 4, 2, 8, 2, 4, 10, 6, 2, 10, 6, 8, 12, 6, 2, 18, 6, 8, 16, 8, 8, 24, 2, 6, 20, 14, 12, 26, 6, 6, 24, 22, 6, 30, 6, 20, 30, 10, 8, 36, 14, 18, 32, 18, 4, 48, 18, 22, 30, 14, 12, 52, 14, 20, 42, 26, 24, 44, 6, 20, 52, 38, 12, 54, 10, 26

Offset: 1

Alejandro B. Galván, Dec 29 2023

nonn

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

The number of triangular partitions of size n with any number of removable cells is in A352882.

Cf. A368554 (with two removable cells).

See the g.f. in Proposition 6.2. of Elizalde and Galván.

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

Original entry on oeis.org

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

Alejandro B. Galván, Jan 01 2024

nonn

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

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

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

% 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

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

cp's OEIS Frontend

A368554 a(n) is the number of triangular partitions of size n with two removable cells.

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A368556 a(n) is the number of triangular partitions of size n with just one removable cell.

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

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