Read e-book online Algebraic Combinatorics I: Association Schemes PDF

Combinatorics

By Eiichi Bannai

ISBN-10: 0805304908

ISBN-13: 9780805304909

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22 What Is Enumerative Combinatorics? 1 Proposition. Let v = (a1 , . . , ad ) ∈ Nd , and let ei denote the ith unit coordinate vector in Zd . The number of lattice paths in Zd from the origin (0, 0, . . , 0) d . to v with steps e1 , . . ,a 1 d Proof. Let v0 , v1 , . . , vk be a lattice path being counted. Then the sequence v1 − v0 , v2 − v1 , . . , vk − vk−1 is simply a sequence consisting of ai ei ’s in some order. 22). 1 is the most basic result in the vast subject of lattice path enumeration.

4 Descents In addition to cycle type and inversion table, there is one other fundamental statistic associated with a permutation w ∈ Sn . If w = w1 w2 · · · wn and 1 ≤ i ≤ n − 1, then i is a descent of w if wi > wi+1 , while i is an ascent if wi < wi+1 . ) Define the descent set D(w) of w by D(w) = {i : wi > wi+1 } ⊆ [n − 1]. If S ⊆ [n − 1], then denote by α(S) (or αn (S) if necessary) the number of permutations w ∈ Sn whose descent set is contained in S, and by β(S) (or βn (S)) the number whose descent set is equal to S.

1 Proposition. Let S = {s1 , . . , sk }< ⊆ [n − 1]. Then α(S) = n . s1 , s2 − s1 , s3 − s2 , . . 35) 32 What Is Enumerative Combinatorics? Proof. To obtain a permutation w = w1 w2 · · · wn ∈ Sn satisfying D(w) ⊆ S, first choose w1 < w2 < · · · < ws1 in sn ways. Then choose ws1 +1 < ws1 +2 < · · · < ws2 in n−s1 s2 −s1 1 ways, and so on. We therefore obtain α(S) = = n s1 n − s1 s2 − s1 n − s2 n − sk ··· s3 − s2 n − sk n , s1 , s2 − s1 , s3 − s2 , . . , n − sk as desired. 2 Example. Let n ≥ 9. Then βn (3, 8) = αn (3, 8) − αn (3) − αn (8) + αn (∅) = n n n − − + 1.

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Algebraic Combinatorics I: Association Schemes by Eiichi Bannai


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