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

Federico Castillo and Jean-Philippe LabbĂ©, 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

KEYWORD

AUTHOR

Colin Mallows, Feb 08 2013

EXTENSIONS

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

STATUS

approved