A356114 -id:A356114 - cp's OEIS frontend

A356263 Triangle read by rows. The reduced triangle of the partition triangle of irreducible permutations (A356262). T(n, k) for n >= 1 and 0 <= k < n.

1, 0, 1, 0, 2, 1, 0, 3, 9, 1, 0, 5, 41, 24, 1, 0, 8, 150, 247, 55, 1, 0, 14, 494, 1746, 1074, 118, 1, 0, 24, 1537, 10126, 13110, 4050, 245, 1, 0, 43, 4642, 52129, 122521, 79396, 14111, 500, 1, 0, 77, 13745, 248494, 967644, 1126049, 425471, 46833, 1011, 1

Offset: 1

Peter Luschny, Aug 01 2022

The triangle can be seen as Euler's triangle A008292 restricted to irreducible permutations.

See the comments in A356116 for the definition of the terms 'partition triangle' and 'reduced partition triangle'. The reduction procedure is formalized in the Sage program in A356116.

			[1] [1]
[2] [0,  1]
[3] [0,  2,     1]
[4] [0,  3,     9,      1]
[5] [0,  5,    41,     24,      1]
[6] [0,  8,   150,    247,     55,       1]
[7] [0, 14,   494,   1746,   1074,     118,     1]
[8] [0, 24,  1537,  10126,  13110,    4050,   245,      1]
[9] [0, 43,  4642,  52129, 122521,   79396, 14111,    500,    1]
[10][0, 77, 13745, 248494, 967644, 1126049, 425471, 46833, 1011, 1]
.
The 5 irreducible permutations counted with T(5, 2) are 23451, 51234, 31524, 34512, and 45123.

Peter Luschny, Permutations with Lehmer, a SageMath Jupyter Notebook.

Cf. A356262 (partition triangle), A007059 (column 2), A003319 (row sums), A356114 (subdiagonal).

Cf. A008292, A356116.

# Uses function 'reduce_partition_triangle' from A356116.
reduce_partition_triangle(A356262_row, 8)

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A356263 Triangle read by rows. The reduced triangle of the partition triangle of irreducible permutations (A356262). T(n, k) for n >= 1 and 0 <= k < n.

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