The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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

%I

%S 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,

%T 2583,132,1,1,429,23580

%N 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).

%C 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.

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

%C The original version of this sequence was

%C 1 1 1 1 1 1 1 ...

%C 1 2 5 14 42 132 428 ...

%C 1 5 24 77 ...

%C 1 14 77 ...

%C 1 42 ...

%C ...

%C but some of the later entries seem to be incorrect. - _Robert J. Vanderbei_, Jan 09 2015

%H C. Mallows, R. J. Vanderbei, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Vanderbei/vand3.html">Which Young Tableaux Can Represent an Outer Sum?</a>, J. Int. Seq. 18 (2015) #15.9.1.

%H Robert J. Vanderbei, <a href="/A211400/a211400.txt">Solutions for the 3 X 3 case</a>

%H Robert J. Vanderbei, <a href="/A211400/a211400_1.txt">Solutions for the 3 X 4 case</a>

%H Robert J. Vanderbei, <a href="/A211400/a211400_2.txt">Solutions for the 4 X 4 case</a>

%e The vectors a = (0,2) and b = (0,4,5) give the outer sums

%e 0 4 5 which have ranks 1 3 4

%e 2 6 7 2 5 6

%e which is one of the five 2 X 3 Young tableaux.

%e One of the 18 3 X 3 tableaux that cannot be realized as a set of outer sums

%e is 1 2 6

%e 3 5 7

%e 4 8 9.

%e The array begins

%e 1 1 1 1 1 1 1 1 1 ...

%e 1 2 5 14 42 132 429 1430 4862 ... (A000108)

%e 1 5 36 295 2583 23580 221680 ... (A255489)

%e 1 14 295 6660 ...

%e 1 42 2583 ...

%e 1 132 23580 ...

%e 1 429 221680 ...

%e 1 1430 ...

%e 1 4862 ...

%e ...

%Y Cf. A060854, A000108, A255489.

%K nonn,hard,more,tabl

%O 1,5

%A _Colin Mallows_, Feb 08 2013

%E Corrected and extended by _Robert J. Vanderbei_, Jan 09 2015

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified January 25 19:09 EST 2020. Contains 331249 sequences. (Running on oeis4.)