OFFSET
0,2
COMMENTS
a(n) is bounded above by 2^(n^2) and bounded below by A048163(n + 1).
Also the number of acyclic edge sets of the complete bipartite graph K_{n,n}. See proof by David E. Speyer at the Mathematics Stack Exchange link below.
It is also the number of n X n binary matrices that can be completed to the all-ones matrix by repeatedly changing an element from a zero to a one if that element is the only zero in its row or column. (Proof idea: Every acyclic edge set can be reduced to the empty set by removing one leaf edge at a time.) This can be interpreted as the number of ways of turning off storage nodes in an n X n array so that data remains restorable in the "full scheme" RAID (Redundant Array of Inexpensive Disks) design described by Nagel, Süß.- Jair Taylor, Jul 29 2019
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..100
Mathematics Stack Exchange, Spanning forests of bipartite graphs and distinct row/column sums of binary matrices
Lars Nagel & Tim Süß, Computing the Probability for Data Loss in Two-Dimensional Parity RAIDs, 2017 13th European Dependable Computing Conference (EDCC), 58-65.
Rebecca J. Stones, Computing the number of h-edge spanning forests in complete bipartite graphs, Discrete Mathematics & Theoretical Computer Science, vol. 16, no. 1, pp. 313-326, 2014.
EXAMPLE
For n = 3, both of the following 3 X 3 (0,1)-matrices have identical row and column sums:
[1 0 1] [1 1 0]
[0 1 0] and [0 1 0]
[0 1 0] [0 0 1]
have row sums of [2 1 1] and column sums of [1 2 1].
So ([2 1 1], [1 2 1]) is one of the 328 possible row/column sums for a 3 X 3 matrix.
MATHEMATICA
{1}~Join~Array[Length@ Union@ Map[Total /@ {Transpose@ #, #} &, Tuples[{0, 1}, {#, #}]] &, 4] (* Michael De Vlieger, Jan 11 2018 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Peter Kagey, Dec 25 2017
EXTENSIONS
Terms a(6) and beyond from Andrew Howroyd, Oct 29 2019
STATUS
approved