By Martin Aigner
Combinatorial enumeration is a quite simply available topic filled with simply acknowledged, yet occasionally tantalizingly tricky difficulties. This e-book leads the reader in a leisurely manner from the fundamental notions to quite a few 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 data for the pro mathematician who desires to research extra concerning the topic. The publication is equipped in 3 components: fundamentals, tools, and themes. There are 666 workouts, and as a different characteristic each bankruptcy ends with a spotlight, discussing a very appealing or recognized result.
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Additional resources for A Course in Enumeration
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. Cambridge Univ. Press, Cambridge; reprinted in one volume by Chelsea, New York, 1960.
To s ∈ S(k1 , . . , 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 !
K k=0 In conclusion, we have found a formula that combines powers, falling factorials, and Stirling numbers, and our familiar polynomial argument gives the polynomial identity n xn = Sn,k x k . (2) k=0 Note that we may stop the summation at n, since obviously Sn,k = 0 for n < k. Polynomial Sequences. Formula (2) is the ﬁrst instance in which two polynomial sequences (x n ) and (x n ) are linearly connected. Such “connecting” identities are a fertile source for various relations involving combinatorial coeﬃcients.
A Course in Enumeration by Martin Aigner