Presentation Title

Once Upon a Row, Once Upon a Column

Format of Presentation

Poster to be presented the Friday of the conference

Abstract

A “Latin square of order n” is an “n by n” array of the symbols 1, 2, ... , n, such that each symbol appears exactly once in each row and once in each column, as in a Sudoku puzzle. The rows and columns of a Latin square correspond to permutations of the set {1, 2, ... , n}, and can each be given a “sign”: +1 if the corresponding permutation is generated by an even number of transpositions (exchanges of two elements), or -1 if the permutation is generated by an odd number of transpositions. The “sign” of a Latin square is then defined to be the product of the signs of its rows and columns. Alon and Tarsi conjectured that if n is an even positive integer, then the sum of the signs of the Latin squares of order n is non-zero, proven for certain special values of n: Drisko proved the conjecture when n is one more than an odd prime; and Glynn proved it when n is one less than an odd prime. Alon and Tarsi’s conjecture is related to several others in combinatorics. For example, Onn showed that it implies Rota’s basis conjecture, whose proof is a major unsolved problem in matroid theory. In our research, we are searching for new approaches to proving Alon and Tarsi’s conjecture. Our strategy is to discover connections and extensions of previous work on this problem.

Department

Mathematics and Statistics

Faculty Advisor

Sean McGuinness

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Once Upon a Row, Once Upon a Column

A “Latin square of order n” is an “n by n” array of the symbols 1, 2, ... , n, such that each symbol appears exactly once in each row and once in each column, as in a Sudoku puzzle. The rows and columns of a Latin square correspond to permutations of the set {1, 2, ... , n}, and can each be given a “sign”: +1 if the corresponding permutation is generated by an even number of transpositions (exchanges of two elements), or -1 if the permutation is generated by an odd number of transpositions. The “sign” of a Latin square is then defined to be the product of the signs of its rows and columns. Alon and Tarsi conjectured that if n is an even positive integer, then the sum of the signs of the Latin squares of order n is non-zero, proven for certain special values of n: Drisko proved the conjecture when n is one more than an odd prime; and Glynn proved it when n is one less than an odd prime. Alon and Tarsi’s conjecture is related to several others in combinatorics. For example, Onn showed that it implies Rota’s basis conjecture, whose proof is a major unsolved problem in matroid theory. In our research, we are searching for new approaches to proving Alon and Tarsi’s conjecture. Our strategy is to discover connections and extensions of previous work on this problem.