# Which young tableaux can represent an outer sum?

## Author(s): Mallows, C; Vanderbei, Robert J.

DownloadTo refer to this page use:

`http://arks.princeton.edu/ark:/88435/pr14p4b`

Abstract: | © 2015, University of Waterloo. All rights reserved.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. |

Publication Date: | 30-Jul-2015 |

Citation: | Mallows, C, Vanderbei, RJ. "Which young tableaux can represent an outer sum?" Journal of Integer Sequences, 18(9), 2015. |

EISSN: | 1530-7638 |

Type of Material: | Journal Article |

Journal/Proceeding Title: | Journal of Integer Sequences |

Version: | This is the author’s final manuscript. All rights reserved to author(s). |

Items in OAR@Princeton are protected by copyright, with all rights reserved, unless otherwise indicated.