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%I #13 Sep 11 2020 23:47:14
%S 1,1,1,1,1,1,1,2,2,1,1,3,1,3,1,1,2,3,3,2,1,1,4,2,3,2,4,1,1,2,4,3,3,4,
%T 2,1,1,4,2,9,1,9,2,4,1,1,3,4,3,4,4,3,4,3,1,1,4,3,10,2,7,2,10,3,4,1,1,
%U 2,4,6,4,4,4,4,6,4,2,1,1,6,2,9,3,16,1,16,3,9,2,6,1,1,2,6,3,4,9,4,4,9,4,3,6,2,1
%N Array read by antidiagonals: T(m,n) (m>=1, n>=1) is number of alpha-labelings of the complete bipartite graph K_{m,n}.
%C (We continue from the comments in A337278)
%C An alpha-labeling of a bipartite graph is a graceful labeling with the further proviso that all labels of one part are less than all labels of the other part. In particular, K_{3,4} has only 3 alpha-labelings, because the third example shown in the comments in A337278 isn't alpha.
%H Joseph A. Gallian, <a href="https://www.combinatorics.org/ds6">Graph Labeling</a>, Electron. J. Combin., Dynamic Survey 6.
%H Don Knuth, <a href="http://cs.stanford.edu/~knuth/programs/back-graceful-kmn.w">CWEB program to generate solutions</a>
%e The array begins:
%e ..1..1..1..1..1..1..1..1..1..1..1..1..1..1..1..1,...
%e ..1..1..2..3..2..4..2..4..3..4..2..6..2..4..4..5,...
%e ..1..2..1..3..2..4..2..4..3..4..2..6..2..4..4..5,...
%e ..1..3..3..3..3..9..3.10..6..9..3.18..3..9..9.15,...
%e ..1..2..2..3..1..4..2..4..3..4..2..6..2..4..4..5,...
%e ..1..4..4..9..4..7..4.16..9.14..4.30..4.14.14.25,...
%e ..1..2..2..3..2..4..1..4..3..4..2..6..2..4..4..5,...
%e ..1..4..4.10..4.16..4.10.10.16..4.40..4.16.16.35,...
%e ..1..3..3..6..3..9..3.10..3..9..3.18..3..9..9.15,...
%e ..1..4..4..9..4.14..4.16..9..7..4.30..4.14.14.25,...
%e ..1..2..2..3..2..4..2..4..3..4..1..6..2..4..4..5,...
%e ..1..6..6.18..6.30..6.40.18.30..6.42..6.30.30.75,...
%e ..1..2..2..3..2..4..2..4..3..4..2..6..1..4..4..5,...
%e ..1..4..4..9..4.14..4.16..9.14..4.30..4..7.14.25,...
%e ..1..4..4..9..4.14..4.16..9.14..4.30..4.14..7.25,...
%e ..1..5..5.15..5.25..5.35.15.25..5.75..5.25.25..?,...
%e ...
%e The first few antidiagonals are:
%e 1,
%e 1,1,
%e 1,1,1,
%e 1,2,2,1,
%e 1,3,1,3,1,
%e 1,2,3,3,2,1,
%e 1,4,2,3,2,4,1,
%e 1,2,4,3,3,4,2,1,
%e 1,4,2,9,1,9,2,4,1,
%e 1,3,4,3,4,4,3,4,3,1
%e ...
%Y Cf. A337278.
%K nonn,tabl
%O 1,8
%A _N. J. A. Sloane_, Sep 11 2020, based on a communication from _Don Knuth_, Sep 08 2020