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

1,5

Comments

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.

The entries at T(2,n) and T(m,2) are Catalan numbers (A000108).

The original version of this sequence was

1 1 1 1 1 1 1 ...

1 2 5 14 42 132 428 ...

1 5 24 77 ...

1 14 77 ...

1 42 ...

...

but some of the later entries seem to be incorrect. - Robert J. Vanderbei, Jan 09 2015

Links

Table of n, a(n) for n=1..31.

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

Example

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   ...
...

Crossrefs

Cf. A060854, A000108, A255489.

Sequence in context: A306344 A128612 A284731 * A060854 A091378 A156045

Adjacent sequences: A211397 A211398 A211399 * A211401 A211402 A211403

Keyword

nonn,hard,more,tabl

Author

Colin Mallows, Feb 08 2013

Extensions

Corrected and extended by Robert J. Vanderbei, Jan 09 2015

Status

approved