

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



