A297077 Number of distinct row and column sums for n X n (0, 1)-matrices.

1, 2, 15, 328, 16145, 1475856, 219682183, 48658878080, 15047189968833, 6199170628499200, 3283463201858585471, 2174417430787021427712, 1760550428764505109190225, 1711145965399957937819322368, 1966168979042910307305159432375, 2636533865690624357354875499216896

Offset: 0

Author

Peter Kagey, Dec 25 2017

Keywords

• nonn

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

Examples

			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.
		

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.

Crossrefs

Main diagonal of A328887.

Cf. A048163 gives the number of row/column sums that uniquely identify a matrix.

Programs

Mathematica

{1}~Join~Array[Length@ Union@ Map[Total /@ {Transpose@ #, #} &, Tuples[{0, 1}, {#, #}]] &, 4] (* Michael De Vlieger, Jan 11 2018 *)

Extensions

Terms a(6) and beyond from Andrew Howroyd, Oct 29 2019