A096121 -id:A096121 - cp's OEIS frontend

A269565 Array read by antidiagonals: T(n,m) is the number of (directed) Hamiltonian paths in K_n X K_m.

1, 2, 2, 6, 8, 6, 24, 60, 60, 24, 120, 816, 1512, 816, 120, 720, 17520, 83520, 83520, 17520, 720, 5040, 550080, 8869680, 22394880, 8869680, 550080, 5040, 40320, 23839200, 1621680480, 13346910720, 13346910720, 1621680480, 23839200, 40320

Offset: 1

Andrew Howroyd, Feb 29 2016

Equivalently, the number of directed Hamiltonian paths on the n X m rook graph.
Conjecture: T(n,m) mod n!*m! = 0. - Mikhail Kurkov, Feb 08 2019
The above conjecture is true since a path defines an ordering on the rows and columns by the order in which they are first visited by the path. Every permutation of rows and columns therefore gives a different path. - Andrew Howroyd, Feb 08 2021

			Array begins:
===========================================================
n\m|    1      2        3            4               5
---+-------------------------------------------------------
1  |    1,     2,       6,          24,            120, ...
2  |    2,     8,      60,         816,          17520, ...
3  |    6,    60,    1512,       83520,        8869680, ...
4  |   24,   816,   83520,    22394880,    13346910720, ...
5  |  120, 17520, 8869680, 13346910720, 50657369241600, ...
...

Andrew Howroyd, Table of n, a(n) for n = 1..96
Eric Weisstein's World of Mathematics, Hamiltonian Path
Eric Weisstein's World of Mathematics, Rook Graph

Main diagonal is A096970.
Columns 2..3 are A096121, A329319.
Cf. A286418, A269562.

From Andrew Howroyd, Oct 20 2019: (Start)
T(n,m) = T(m,n).
T(n,1) = n!. (End)

A329319 Number of (directed) Hamiltonian paths in K_{n} X K_{3}.

Original entry on oeis.org

6, 60, 1512, 83520, 8869680, 1621680480, 472907393280, 207307564531200, 130417226086775040, 113438068529746060800, 132325125941706622848000, 201817805274824171102208000

Offset: 1

Mikhail Kurkov, Nov 10 2019

Equivalently, number of full spectrum rook's walks on a (n X 3) board.

Column 3 of A269565.

Cf. A096121.

A360878 Number of (undirected) paths in the 2 X n rook graph.

Original entry on oeis.org

1, 12, 129, 1984, 45945, 1524156, 68838217, 4070403744, 305642504529, 28440008182540, 3214141725643761, 433856895597946272, 68964321078341276809, 12753724616472980432124, 2715405762438952565521785, 659549661987730244294458816, 181293528280954206831103494177

Offset: 1

Andrew Howroyd, Feb 25 2023

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..100

Row 2 of A360877.

Cf. A096121, A276356, A341500.

a(n)={sum(k=2, n, binomial(n,k)*k!) + sum(k=1, n, k*binomial(n,k)*binomial(k-1,k\2)*sum(i=0, n-k, binomial(n-k,i)*(k\2+i)!)*sum(i=0, n-k, binomial(n-k,i)*((k-1)\2+i)!))} \\ Andrew Howroyd, May 28 2025

a(n) = (Sum_{k=2..n} binomial(n,k)*k!) + (Sum_{k=1..n} k*binomial(n,k)*binomial(k-1, floor(k/2)) * (Sum_{i=0..n-k} binomial(n-k,i)*(floor(k/2)+i)!) * (Sum_{i=0..n-k} binomial(n-k,i)*(floor((k-1)/2)+i)!)). - Andrew Howroyd, May 28 2025

a(8) onwards from Andrew Howroyd, May 28 2025

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A269565 Array read by antidiagonals: T(n,m) is the number of (directed) Hamiltonian paths in K_n X K_m.

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A329319 Number of (directed) Hamiltonian paths in K_{n} X K_{3}.

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A360878 Number of (undirected) paths in the 2 X n rook graph.

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