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A216678 On an n X n grid, number of ways to draw arrows between adjacent nodes such that each node has one outgoing and one incoming arrow, of which the one is not the opposite of the other (i.e., without 2-loops). 3
0, 2, 0, 88, 0, 207408, 0, 22902801416, 0, 112398351350823112, 0, 24075116871728596710774372 (list; graph; refs; listen; history; text; internal format)



"Adjacent" is meant in the sense of von Neumann neighborhoods (4 neighbors for "interior" nodes, 3 resp. 2 for nodes on the borders resp. in the corners).

Or: Number of permutations of an n X n array, with each element moving exactly one horizontally or vertically and without 2-loops.


Table of n, a(n) for n=1..12.

Project Euler, Problem 393: Migrating ants.


For a 1 X 1 grid, there is no such permutation or possibility.

For a 2 X 2 grid, on has the clockwise and counterclockwise cyclic "permutation" of the 4 nodes. (It is not allowed to draw arrows between 2 pairs of nodes in horizontal or vertical sense since, e.g., the arrow from the first to the second node is the opposite of the arrow from the second to the first node.)

For a 3 X 3 grid, there is no possibility, neither for a 5 X 5 grid.


See A216675 for the same problem without the additional restriction.

Cf. A216796, A216797, A216798, A216799, A216800 for more general n X k grids.

Sequence in context: A012447 A136558 A156490 * A136559 A009740 A132860

Adjacent sequences:  A216675 A216676 A216677 * A216679 A216680 A216681




M. F. Hasler, Sep 13 2012


Terms beyond a(5) computed by R. H. Hardin, Sep 15 2012



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Last modified May 24 18:34 EDT 2019. Contains 323534 sequences. (Running on oeis4.)