A357760 -id:A357760 - cp's OEIS frontend

A358481 a(n) is the number of different pairs of shortest grid paths joining two opposite corners in opposite order in an n X n X n grid without having middle point on their paths as a common point.

30, 6218, 2658432, 1054788750, 552306591900, 269380692717696, 155175092086118400, 83798883891736779150, 50885239237727996887500, 29198209396114625497699068, 18332853214682572877389897728, 10951674446687597386319569942656, 7036938452279110885561897815723264, 4325988198220149508865311059521280000

Offset: 1

Janaka Rodrigo, Nov 18 2022

nonn

Alternatively a(n) is the number of ways two ants can interchange their positions starting simultaneously from two opposite corners and moving along shortest grid paths at same speed in an n X n X n grid without meeting other one.

			When n=2 number of ways to move between two opposite corners are given by 6!/(2!*2!*2!) and number of such pairs are given by (6!/(2!*2!*2!))^2. This total number of pairs are given by A268553(2)=8100.
Number of pairs which have the middle point of their paths as a common point are A357760(2)=1782.
Therefore number of pairs without having middle point on their paths as a common point are 8100-1782=6218

Cf. A268553, A357760.

a(n) = A268553(n) - A357760(n).

A360444 a(n) is the number of ways for two nonintersecting, unordered pairs of shortest grid paths to cross over between two opposite corners in an n X n grid without intersecting opposite paths at their middle points.

Original entry on oeis.org

0, 0, 52, 4540, 742404, 103625004, 16451015760, 2693403573732, 463439672732740, 82516389937797244, 15153421065014201424, 2855078861978328905660, 550005952989718178915472, 108007620360200608500699120, 21569526154939330279935568704

Offset: 1

Janaka Rodrigo, Jul 15 2023

nonn

Alternatively, the number of different ways when two of four ants start at one corner of the grid and other two start at the opposite corner at the same time t and they all stop moving at time T (at which time each ant reaches the corner opposite from its starting corner) and at no time in the open interval (t,T) does any ant meet any other ant.

			In the 1 X 1 and 2 X 2 grids there is no possibility of this happening.
In a 3 X 3 grid, if a pair starting from the bottom left corner move along NNNEEE and EEENNN, the pair starting from the top right corner can move along WWSWSS and SSWSWW (this is only one of the nine options available for the second pair) so that they can cross over without meeting any other. There are 52 different ways to do this.

Cf. A362207, A358481, A357760, A357603, A000891.

cp's OEIS Frontend

A358481 a(n) is the number of different pairs of shortest grid paths joining two opposite corners in opposite order in an n X n X n grid without having middle point on their paths as a common point.

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