Enumerative and Algebraic Combinatorics

Enumeration, alias counting, is the oldest mathematical subject, while Algebraic Combinatorics is one of the youngest. Some cynics claim that Algebraic Combinatorics is not really a new subject but just a new name given to Enumerative Combinatorics in order to enhance its (former) poor image, but Algebraic Combinatorics is in fact the synthesis of two opposing trends: abstraction of the concrete and concretization of the abstract. The former trend dominated the first half of the 20th century, starting with Hilbert’s ‘theological’ proof of the fundamental theorem of invariants, while the latter trend is dominating contemporary mathematics, thanks in large part to Its Omnipresence, The Mighty Computer. The abstraction trend, that consists of categorization, conceptualization, structuralization and fancification (in short ‘Bourbakisation’) of mathematics, did not escape enumeration, and in the hands of such giants as Gian-Carlo Rota and Richard Stanley in America and Marco Schützenberger and Dominique Foata in France, classical, enumerative, combinatorics became more conceptual, structural and algebraic. On the other hand, the trend towards the explicit, concrete, and constructive revealed that many algebraic structures have hidden combinatorial underpinnings, and the attempts to unearth them lead to the other route towards the establishment of Algebraic Combinatorics as a full-fledged separate mathematical specialty. Enumeration The Fundamental Theorem of Enumeration, independently discovered by several anonymous cave-dwellers, states that