%I
%S 1,1,1,1,2,1,1,5,5,1,1,14,36,14,1,1,42,295,295,42,1,1,132,2583,6660,
%T 2583,132,1,1,429,23580
%N Rectangular array, read by upward diagonals: T(n,m) is the number of Young tableaux that can be realized as the ranks of the outer sums a_i + b_j where a = (a_1, ... a_n) and b = (b_1, ... b_m) are real monotone vectors in general position (all sums different).
%C Alternatively, that can be realized as the ranks of the outer products a_i b_j where a = (a_1, ... a_n) and b = (b_1, ... b_m) are real positive monotone vectors.
%C The entries at T(2,n) and T(m,2) are Catalan numbers (A000108).
%C The original version of this sequence was
%C 1 1 1 1 1 1 1 ...
%C 1 2 5 14 42 132 428 ...
%C 1 5 24 77 ...
%C 1 14 77 ...
%C 1 42 ...
%C ...
%C but some of the later entries seem to be incorrect.  _Robert J. Vanderbei_, Jan 09 2015
%H C. Mallows, R. J. Vanderbei, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Vanderbei/vand3.html">Which Young Tableaux Can Represent an Outer Sum?</a>, J. Int. Seq. 18 (2015) #15.9.1.
%H Robert J. Vanderbei, <a href="/A211400/a211400.txt">Solutions for the 3 X 3 case</a>
%H Robert J. Vanderbei, <a href="/A211400/a211400_1.txt">Solutions for the 3 X 4 case</a>
%H Robert J. Vanderbei, <a href="/A211400/a211400_2.txt">Solutions for the 4 X 4 case</a>
%e The vectors a = (0,2) and b = (0,4,5) give the outer sums
%e 0 4 5 which have ranks 1 3 4
%e 2 6 7 2 5 6
%e which is one of the five 2 X 3 Young tableaux.
%e One of the 18 3 X 3 tableaux that cannot be realized as a set of outer sums
%e is 1 2 6
%e 3 5 7
%e 4 8 9.
%e The array begins
%e 1 1 1 1 1 1 1 1 1 ...
%e 1 2 5 14 42 132 429 1430 4862 ... (A000108)
%e 1 5 36 295 2583 23580 221680 ... (A255489)
%e 1 14 295 6660 ...
%e 1 42 2583 ...
%e 1 132 23580 ...
%e 1 429 221680 ...
%e 1 1430 ...
%e 1 4862 ...
%e ...
%Y Cf. A060854, A000108, A255489.
%K nonn,hard,more,tabl
%O 1,5
%A _Colin Mallows_, Feb 08 2013
%E Corrected and extended by _Robert J. Vanderbei_, Jan 09 2015
