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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). 2
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 (list; table; graph; refs; listen; history; text; internal format)
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.

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

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Last modified December 8 02:30 EST 2019. Contains 329850 sequences. (Running on oeis4.)