By Martin Aigner
Combinatorial enumeration is a without problems available topic choked with simply said, yet occasionally tantalizingly tricky difficulties. This e-book leads the reader in a leisurely approach from the elemental notions to a number of themes, starting from algebra to statistical physics. Its objective is to introduce the coed to a fascinating box, and to be a resource of knowledge for the pro mathematician who desires to examine extra concerning the topic. The publication is geared up in 3 components: fundamentals, tools, and issues. There are 666 workouts, and as a different function each bankruptcy ends with a spotlight, discussing a very attractive or recognized result.
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This is the desired bijection, also for the second equality. Lattice Paths. To end these introductory remarks about number-partitions we discuss an important and perhaps unexpected connection to binomial counts the latcoeﬃcients via lattice paths. Remember that m+n m tice paths from (0, 0) to (m, n). The following ﬁgure demonstrates a natural bijection between these paths and the set Par( ; ≤ n; ≤ m), by interpreting the part above the lattice path as a Ferrers diagram. Example. (5, 4) n=4 (0, 0) → λ = 4311 m=5 Hence p( ; ≤ n; ≤ m) = m+n .
N − k]q ! (n ≥ k ≥ 0) . (3) This is certainly true for n = 0 or k = 0; we proceed by induction 1−qn on n. From [n]q ! = 1−q [n − 1]q ! and (1) we obtain n k q n−1 n−1 + qk q k−1 q k [n − 1]q ! [n − 1]q ! [n − k]q ! [n − 1 − k]q ! = = [n − 1]q ! [n − k]q ! = [n − 1]q ! [n]q ! 1 − qn · = . [n − k]q ! [n − k]q ! n 1 − qk qk (1 − qn−k ) + 1−q 1−q n Note that (3) implies [ k ]q = [ n−k ]q . Cancelling terms we may also write 38 1 Fundamental Coeﬃcients [n]q [n − 1]q · · · [n − k + 1]q (1 − qn ) · · · (1 − qn−k+1 ) = , q [k]q !
Usually, we assume that all pn (x) have leading coeﬃcient 1. Any such sequence pn (x) is a basis of the vector space K[x]. Hence, given two such sequences pn (x) and qn (x) there are unique coeﬃcients an,k and bn,k with n pn (x) = n an,k qk (x), qn (x) = k=0 bn,k pk (x) . (3) k=0 The numbers an,k respectively bn,k are called the connecting coeﬃcients between the sequences pn (x) and qn (x) . They form inﬁnite lower triangular matrices (an,k ) and (bn,k ), since clearly an,k = bn,k = 0 for n < k.