 Journal of Integer Sequences, Vol. 18 (2015), Article 15.9.1

## Which Young Tableaux Can Represent an Outer Sum?

### Colin Mallows Flemington, NJ USA Robert J. Vanderbei Department of Operations Research and Financial Engineering Princeton University Princeton, NJ 08544 USA

Abstract:

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.

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(Concerned with sequences A000108 A211400.)

Received April 1 2015; revised versions received July 23 2015; July 30 2015. Published in Journal of Integer Sequences, July 30 2015.