By Martin Aigner

Combinatorial enumeration is a effectively available topic choked with simply acknowledged, yet occasionally tantalizingly tough difficulties. This publication leads the reader in a leisurely approach from the elemental notions to a number of subject matters, starting from algebra to statistical physics. Its goal is to introduce the coed to a fascinating box, and to be a resource of knowledge for the pro mathematician who desires to research extra concerning the topic. The publication is geared up in 3 elements: fundamentals, tools, and themes. There are 666 workouts, and as a different function each bankruptcy ends with a spotlight, discussing a very appealing or recognized result.

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Km ) we associate a pair (t, u) of words such that t ∈ S(k1 , . . , km−2 , km−1 + km ) arises from s by replacing in s every m by m − 1, and where u ∈ S(km−1 , km ) is the subword of m − 1 and m. Example. s = 21331211 → t = 21221211, u = 2332. Since s can be recovered uniquely from t and u, the map s (t, u) is a bijection. Furthermore, we have inv(s) = inv(t) + inv(u), that is, s qinv(s) = t qinv(t) · u qinv(u) . With induction this gives qinv(t) = t [n]q ! [k1 ]q ! [km−1 + km ]q ! qinv(u) = u [km−1 + km ]q !

L. Comtet (1974): Advanced Combinatorics. Reidel, Dordrecht and Boston. 3. N. Elkies, G. Kuperberg, M. Larsen, and J. Propp (1992): Alternating sign matrices and domino tilings, parts I and II. Journal Algebr. Comb. 1, 111–132, 219–234. 4. D. Foata and M. Schützenberger (1970): Théorie Géometrique des Polynômes Euleriens. Lecture Notes Math. 138. Springer, Berlin. 5. L. E. Knuth, and O. Patashnik (1994): Concrete Mathematics, 2nd edition. Addison-Wesley, Reading. 6. A. MacMahon (1915): Combinatory Analysis, 2 vols.

Ai . )(aj . ) . , where ai is the ﬁrst entry larger than a1 , aj is the ﬁrst entry after ai larger than ai , and so on. Thus, the number of cycles determined in this way equals the number of left-to-right maxima of the word a1 a2 . . an . A natural and very useful graphical representation is to interpret σ ∈ S(n) as a directed graph with i → j if j = σ (i). The cycles of σ correspond then to the directed circuits of the graph. Example. σ = (142)(738)(5)(69) corresponds to the graph 1 3 6 2 4 7 8 9 5 Deﬁnition.