A Course in Enumeration by Martin Aigner

By Martin Aigner

Combinatorial enumeration is a quite simply obtainable topic jam-packed with simply said, yet occasionally tantalizingly tricky difficulties. This booklet leads the reader in a leisurely manner from the elemental notions to quite a few themes, starting from algebra to statistical physics. Its target is to introduce the scholar to a fascinating box, and to be a resource of knowledge for the pro mathematician who desires to research extra in regards to the topic. The ebook is geared up in 3 components: fundamentals, tools, and themes. There are 666 workouts, and as a distinct function each bankruptcy ends with a spotlight, discussing a very attractive or recognized result.

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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 ! [km ]q ! and we conclude that qinv(s) = s qinv(t) · t qinv(u) = u [n]q ! [k1 ]q ! · · · [km ]q ! n For k1 = k2 = · · · = kn = 1 we have [ 1···1 ]q = and we infer the following corrollary. [n]q !

An , ai = σ (i). The pair {i, j} is called an inversion if i < j but ai > aj . The inversion number inv(σ ) is the number of inversions of σ , and σ is called even (odd) if inv(σ ) is even (odd). The sign of σ is defined as sign (σ ) = (−1)inv(σ ) . Thus sign (σ ) = 1 if σ is even and sign (σ ) = −1 if σ is odd. The following graphic representation of a permutation σ will prove very useful. We write 1, 2, . . , n into two rows and join i with ai = σ (i) by an edge. For example, σ = 314265 has the diagram 1 2 3 4 5 6 1 2 3 4 5 6 28 1 Fundamental Coefficients A moment’s thought shows that {i, j} is an inversion if and only if the edges iai and jaj cross in the diagram.

Bn−1 are already (uniquely) determined. Then n n it follows from 0 = k=0 ak bn−k = a0 bn + k=1 ak bn−k , that n −1 bn = −a0 a b is uniquely determined. k=1 k n−k Example. The geometric series n≥0 zn has as inverse 1 − z by 1 (1); hence we write n≥0 zn = 1−z . Let A(z) = n≥0 an zn . Then by convolution, ⎛ ⎞ n A(z) n n ⎝ = an z · z = ak ⎠ z n , (2) 1 − z n≥0 n≥0 n≥0 k=0 thus A(z) 1−z adds the coefficients up to n. In particular, we have 1 = (n + 1)z n . 1 Algebra of Formal Series zm an z n = n≥0 For example, 55 z (1−z)2 an zn+m = n≥0 = n≥0 nz an−m zn .

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