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).
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, 2583, 132, 1, 1, 429, 23580
Offset: 1
Examples
The vectors a = (0,2) and b = (0,4,5) give the outer sums
0 4 5 which have ranks 1 3 4
2 6 7 2 5 6
which is one of the five 2 X 3 Young tableaux.
One of the 18 3 X 3 tableaux that cannot be realized as a set of outer sums
is 1 2 6
3 5 7
4 8 9.
The array begins
1 1 1 1 1 1 1 1 1 ...
1 2 5 14 42 132 429 1430 4862 ... (A000108)
1 5 36 295 2583 23580 221680 ... (A255489)
1 14 295 6660 ...
1 42 2583 ...
1 132 23580 ...
1 429 221680 ...
1 1430 ...
1 4862 ...
...
Links
- Federico Castillo and Jean-Philippe Labbé, Lineup polytopes of product of simplices, arXiv:2306.00082 [math.CO], 2023.
- C. Mallows, R. J. Vanderbei, Which Young Tableaux Can Represent an Outer Sum?, J. Int. Seq. 18 (2015) #15.9.1.
- Robert J. Vanderbei, Solutions for the 3 X 3 case
- Robert J. Vanderbei, Solutions for the 3 X 4 case
- Robert J. Vanderbei, Solutions for the 4 X 4 case
Extensions
Corrected and extended by Robert J. Vanderbei, Jan 09 2015
Comments