# Algebraic combinatorics by Jürgen Müller

By Jürgen Müller

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In particular, using the constant series associated with a := 1 we get the following: For sequences [fn ∈ K; n ∈ N0 ] and [gn ∈ K; n ∈ N0 ] being related by gn = nk=0 nk fk ∈ K, for all n ∈ N0 , we have g = f · exp ∈ K[[X]], implying f = g · exp(X)−1 = g · exp(−X) ∈ K[[X]], from which we in turn recover the binomial inversion formula fn = nk=0 (−1)n−k nk gk ∈ K, for all n ∈ N0 . , for all n ∈ N0 , hence D · exp = exp(−X) ∈ Q[[X]]; 1−X n (−1)k ∈ Q, for all k=0 k! n! n n≥0 n! X = (1 − X)−1 ∈ Q[[X]], thus D = in particular, from this we recover the formula Dn = n ∈ N0 .

Fixing a (k − 1)-dimensional Fq -subspace V ≤ Fn−1 dimFq (V ∩ Fn−1 q q and going over to Fnq /V , shows that there are q q−1 −1 − q q−1−1 = q n−k subspaces V as above such that V ∩ Fn−1 = V . Thus we get the triangle q n n−1 n−1 n−k · k−1 q , for all n ∈ N and k ∈ {1, . . , n}. Since identity k q = k q + q n−k+1 n−k = 1, for all n ∈ N0 , we see that nk q can be considered as a polynomial with integral coeﬃcients in the indeterminate q, called a Gaussian polynomial. n 0 q i−1 j=0 Since, for i ∈ N, the polynomial function R → R : q → q i − 1 = (q − 1) · has a single zero at q = 1, and we have limq→1 we infer that limq→1 n k q n k q = limq→1 ( k−1 i=0 q −1 q−1 i n−i−1 j q j=0 i j j=0 q = limq→1 ( i−1 j=0 q j ) = i, k−1 n−i n i=0 i+1 = k .

Ii) We have the partial fraction decomposition f = C(X) ∩ C[[X]], where gj ∈ C[X]≤dj −1 for all j ∈ {1, . . , k}. k gj j=1 (1−aj X)dj ∈ III Generating functions 46 iii) The sequence [fn ∈ C; n ∈ N0 ] is a linear recurrent sequence of degree d d, that is fn+d + i=1 qi fn+d−i = 0 for all n ∈ N0 . k iv) We have fn = j=1 hj (n)anj , for all n ∈ N0 , where hj ∈ C[X]≤dj −1 . In order to see this let V(i) , V(ii) , V(iii) , V(iv) ≤ C[[X]] be the C-subspaces of all formal power series fulﬁlling property (i), (ii), (iii) and (iv), respectively.