A335619 Number of fundamentally different graceful labelings of the complete bipartite graph K_{n,n}.

1, 1, 4, 1, 7, 2, 10, 3, 8, 1, 42, 2, 7, 7

Offset

2,3

Links

Table of n, a(n) for n=2..15.

Eric Weisstein's World of Mathematics, Complete Bipartite Graph

Eric Weisstein's World of Mathematics, Graceful Labeling

Formula

a(n) = A337793(n)/(4*(n!)^2).

Example

a(3) = 4 because there are 4 fundamentally different graceful labelings:
solution #1:
   0  1  4  5
   6  8 14 16
solution #2:
   0  1  8  9
  10 12 14 16
solution #3:
   0  1  2 15
   5  9 13 16
solution #4
   0  1  2  3:
   4  8 12 16
All others can be obtained by permutations of left and right vertices, swapping halves ("0" vertex left or right) and the replacement of all vertex labels k by N^2-k. - noted by Bert Dobbelaere, Oct 01 2020

Crossrefs

Cf. A337793 (total number of graceful labelings).

Sequence in context: A050356 A245838 A158860 * A037022 A037023 A143971

Adjacent sequences: A335616 A335617 A335618 * A335620 A335621 A335622

Keyword

nonn,more

Author

Eric W. Weisstein, Oct 02 2020

Extensions

a(10)-a(15) from the diagonal of A337278 by Don Knuth, Dec 08 2020

Status

approved