This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A211400 #41 Jun 14 2023 16:47:53 %S A211400 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 A211400 2583,132,1,1,429,23580 %N A211400 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 A211400 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 A211400 The entries at T(2,n) and T(m,2) are Catalan numbers (A000108). %C A211400 The original version of this sequence was %C A211400 1 1 1 1 1 1 1 ... %C A211400 1 2 5 14 42 132 428 ... %C A211400 1 5 24 77 ... %C A211400 1 14 77 ... %C A211400 1 42 ... %C A211400 ... %C A211400 but some of the later entries seem to be incorrect. - _Robert J. Vanderbei_, Jan 09 2015 %H A211400 Federico Castillo and Jean-Philippe Labbé, <a href="https://arxiv.org/abs/2306.00082">Lineup polytopes of product of simplices</a>, arXiv:2306.00082 [math.CO], 2023. %H A211400 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 A211400 Robert J. Vanderbei, <a href="/A211400/a211400.txt">Solutions for the 3 X 3 case</a> %H A211400 Robert J. Vanderbei, <a href="/A211400/a211400_1.txt">Solutions for the 3 X 4 case</a> %H A211400 Robert J. Vanderbei, <a href="/A211400/a211400_2.txt">Solutions for the 4 X 4 case</a> %e A211400 The vectors a = (0,2) and b = (0,4,5) give the outer sums %e A211400 0 4 5 which have ranks 1 3 4 %e A211400 2 6 7 2 5 6 %e A211400 which is one of the five 2 X 3 Young tableaux. %e A211400 One of the 18 3 X 3 tableaux that cannot be realized as a set of outer sums %e A211400 is 1 2 6 %e A211400 3 5 7 %e A211400 4 8 9. %e A211400 The array begins %e A211400 1 1 1 1 1 1 1 1 1 ... %e A211400 1 2 5 14 42 132 429 1430 4862 ... (A000108) %e A211400 1 5 36 295 2583 23580 221680 ... (A255489) %e A211400 1 14 295 6660 ... %e A211400 1 42 2583 ... %e A211400 1 132 23580 ... %e A211400 1 429 221680 ... %e A211400 1 1430 ... %e A211400 1 4862 ... %e A211400 ... %Y A211400 Cf. A060854, A000108, A255489. %K A211400 nonn,hard,more,tabl %O A211400 1,5 %A A211400 _Colin Mallows_, Feb 08 2013 %E A211400 Corrected and extended by _Robert J. Vanderbei_, Jan 09 2015