## Download Cauchy-Binet by Jim Morrow PDF

By Jim Morrow

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**Example text**

A(xk )) collects all multisets of size k. It is also of interest to consider k-tuples (a1 , a2 , . . , ak ) of diﬀerent elements a1 , a2 , . . , ak ∈ A. There is no general result here as for the unrestricted case. Nevertheless there are some particular cases that can be treated similarly. For example, if C denotes the set of k-tuples (a1 , a2 , . . , ak ) with the property that all aj are diﬀerent we have k c(x) = a(x) − k (−1) =2 k a(x)k− a(x ). l 36 2 Generating Functions This is just a direct application of the principle of inclusion and exclusion.

2 1 2 3 1 3 2! a(x )+t 3! a(x )+··· 1 +2 2 +···+k k =k 1! 2! · · · k ! k! · 1 12 2 · · · k k a(x) 1 a(x2 ) 2 · · · a(xk ) k = PSk (a(x), a(x2 ), . . , a(xk )) collects all multisets of size k. It is also of interest to consider k-tuples (a1 , a2 , . . , ak ) of diﬀerent elements a1 , a2 , . . , ak ∈ A. There is no general result here as for the unrestricted case. Nevertheless there are some particular cases that can be treated similarly. For example, if C denotes the set of k-tuples (a1 , a2 , .

Thus, Fy (x2 , y(x2 )) = 1 and the implicit function theorem shows that there exists an analytic solution y = y(x) in a neighbourhood of x2 . For |x| < x0 this solution equals the power series y(x) and for |x| ≥ x0 it provides an analytic continuation to a region of the form Δ (by compactness it is suﬃcient to consider ﬁnitely many x2 with |x2 | = x0 and | arg(x2 )| ≥ δ). 15). 19, where we add an additional k-dimensional parameter u = (u1 , . . , uk ) (compare also with [59]). This concept turns out to be useful for studying the distribution of tree parameters like the number of leaves.