Geodesic
Which Young tableaux can represent an outer sum?
Journal of integer sequences, Tome 18 (2015) no. 9.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: Given two vectors, not necessarily of the same length, each having increasing elements, we form the matrix whose $(i,j)$-th element is the sum of the $i$-th element from the first vector and the $j$-th element from the second vector. Such a matrix is called an outer sum of the two vectors (a concept that is analogous to outer products). If we assume that all the entries of this matrix are distinct, then we can form another matrix of the same size but for which the $(i,j)$-th element is not the matrix element itself but rather the rank of this element in a sorted list of all the numbers in the first matrix. Such a matrix is called a Young tableau. We say that it "represents" the outer sum. In this paper, we address the question as to whether all Young tableaux can be generated this way. When one of the two dimensions is two, then the answer is yes. In all higher dimensional cases, the answer is no. We prove the positive result and give examples illustrating the negative result.
Classification : 05E05, 05E10, 20C30, 17B10
Keywords: outer sum, Young tableau, Dyck path, Catalan number, linear programming (Concerned with sequences and ) 
@article{JIS_2015__18_9_a3,
     author = {Mallows, Colin and Vanderbei, Robert J.},
     title = {Which {Young} tableaux can represent an outer sum?},
     journal = {Journal of integer sequences},
     publisher = {mathdoc},
     volume = {18},
     number = {9},
     year = {2015},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2015__18_9_a3/}
}
TY  - JOUR
AU  - Mallows, Colin
AU  - Vanderbei, Robert J.
TI  - Which Young tableaux can represent an outer sum?
JO  - Journal of integer sequences
PY  - 2015
VL  - 18
IS  - 9
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/JIS_2015__18_9_a3/
LA  - en
ID  - JIS_2015__18_9_a3
ER  - 
%0 Journal Article
%A Mallows, Colin
%A Vanderbei, Robert J.
%T Which Young tableaux can represent an outer sum?
%J Journal of integer sequences
%D 2015
%V 18
%N 9
%I mathdoc
%U http://geodesic.mathdoc.fr/item/JIS_2015__18_9_a3/
%G en
%F JIS_2015__18_9_a3
Mallows, Colin; Vanderbei, Robert J. Which Young tableaux can represent an outer sum?. Journal of integer sequences, Tome 18 (2015) no. 9. http://geodesic.mathdoc.fr/item/JIS_2015__18_9_a3/