By David M. Bressoud
This creation to fresh advancements in algebraic combinatorics illustrates how study in arithmetic really progresses. the writer recounts the dramatic look for and discovery of an evidence of a counting formulation conjectured within the overdue Nineteen Seventies: the variety of n x n alternating signal matrices, gadgets that generalize permutation matrices. whereas it was once obvious that the conjecture needs to be actual, the facts was once elusive. consequently, researchers turned attracted to this challenge and made connections to points of the invariant conception of Jacobi, Sylvester, Cayley, MacMahon, Schur, and younger; to walls and aircraft walls; to symmetric features; to hypergeometric and easy hypergeometric sequence; and, eventually, to the six-vertex version of statistical mechanics. This quantity is obtainable to an individual with an information of linear algebra, and it comprises wide workouts and Mathematica courses to assist facilitate own exploration. scholars will study what mathematicians really do in an engaging and new quarter of arithmetic, or even researchers in combinatorics will locate whatever distinctive inside Proofs and Confirmations.