|
%I #22 Dec 08 2020 21:59:59
%S 1,1,4,1,7,2,10,3,8,1,42,2,7,7
%N Number of fundamentally different graceful labelings of the complete bipartite graph K_{n,n}.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CompleteBipartiteGraph.html">Complete Bipartite Graph</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GracefulLabeling.html">Graceful Labeling</a>
%F a(n) = A337793(n)/(4*(n!)^2).
%e a(3) = 4 because there are 4 fundamentally different graceful labelings:
%e solution #1:
%e 0 1 4 5
%e 6 8 14 16
%e solution #2:
%e 0 1 8 9
%e 10 12 14 16
%e solution #3:
%e 0 1 2 15
%e 5 9 13 16
%e solution #4
%e 0 1 2 3:
%e 4 8 12 16
%e 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
%Y Cf. A337793 (total number of graceful labelings).
%K nonn,more
%O 2,3
%A _Eric W. Weisstein_, Oct 02 2020
%E a(10)-a(15) from the diagonal of A337278 by _Don Knuth_, Dec 08 2020
| 