Download e-book for kindle: Algebraic Combinatorics by Ulrich Dempwolff

By Ulrich Dempwolff

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Tk ) only over ordered partitions with nontrivial parts and our consideration shows 1 k! ,Sk } f (|S1 |) · · · f (|Sk |) where (T1 , . . , Tk ) ranges over ordered and {S1 , . . , Sk } over unordered partitions of [n]. ,Sk } and set Hk (X) = Hk (X) = n≥1 n≥1 1 n! hk (n) n n! X . ,Sk } f (|S1 |) · · · f (|Sk |)g(k). Then f (|S1 |) · · · f (|Sk |)g(k) X n = g(k) F (X)k . k! Define H0 (X) = g(0). Then G(F (X)) = Hk (X) = g(0) + k≥0 k≥1 = g(0) + k≥1 n≥1 = g(0) + n≥1 1 n! 1 n! k≥1 g(k) F (X)k k!

Tk ) n≥0 fj (n) n n! X . f1 (|T1 |) · · · fk (|Tk |) where (T1 , . . , Tk ) ranges over the ordered partitions of size k of [n] and let n H(X) = n≥0 h(n) n! X be the associated EGF. Then F1 (X) · · · Fk (X) = H(X). 1 by an obvious induction. With the next theorem we compute the composition of EGF’s. 2 (Composition formula for EGF,s) Define for the functions f, g : (n) n n N → C, f (0) = 0, the EGF’s F (X) = n≥0 f n! X , G(X) = n≥0 g(n) n! X . ,Sk } f (|S1 |) · · · f (|Sk |) g(k), n ≥ 1, where {S1 , .

An approach with particular emphasis on combinatorial requirements is given by Tutte, On elementary calculus and the Good formula, J. Comb. Theory (B) 18(1975), 97-137. This article is somewhat difficult to read as the author uses a rather unconventional notation. Another useful survey article is Niven, Formal power series, Amer. Math. Month. 76(1969), 871-889. 6 Generating Functions and Recurrences The theme of this section is to turn number sequences into formal power series. Then one manipulates the power series to obtain more information about the number sequence.

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Algebraic Combinatorics by Ulrich Dempwolff

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