By Richard P. Stanley

Catalan numbers are the most ubiquitous series of numbers in arithmetic. This e-book presents, for the 1st time, a complete number of their houses and purposes in combinatorics, algebra, research, quantity idea, chance thought, geometry, topology, and different parts. After an creation to the fundamental houses of Catalan numbers, the ebook offers 214 other forms of gadgets that are counted utilizing Catalan numbers, together with of routines with ideas. The reader can try out fixing the workouts or just flick thru them. sixty eight extra workouts with prescribed hassle degrees current numerous houses of Catalan numbers and similar numbers, akin to Fuss-Catalan numbers, Motzkin numbers, Schröder numbers, Narayana numbers, tremendous Catalan numbers, q-Catalan numbers and (q,t)-Catalan numbers. The e-book concludes with a background of Catalan numbers by way of Igor Pak and a thesaurus of keywords. no matter if your curiosity in arithmetic is game or study, you will discover lots of interesting and stimulating proof right here.

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Note also that when all endpoints are removed (together with the incident edges), we obtain the trees of item 4. 7. When the root is removed, we obtain the trees of item 5. See also Klarner [32]. 8. 1 gives a bijection between the present problem and item 26. An elegant bijection with item 6 was given by F. Bernhardt, private communication, 1996. 9. Traverse the tree in preorder. , away from the root), record 1 if this edge goes to the left or straight down, and record −1 if this edge goes to the right.

5). (a1 , a2 ) ∅ (a1 , a4 ) (a3 , a4 ) (a1 , a2 , a3 , a4 ) 101. Sequences a1 , a2 , . . , a2n−2 of n−1 1’s and n−1 −1’s such that if ai = −1 then either ai+1 = ai+2 = · · · = a2n−2 = −1 or ai+1 + ai+2 + · · · + ai+j > 0 for some j ≥ 1. 1, 1, −1, −1 1, −1, 1, −1 − 1, 1, 1, −1 − 1, 1, −1, 1 − 1, −1, 1, 1 102. Sequences a1 , a2 , . . , an of integers such that a1 = 1, an = ±1, ai = 0, and ai+1 ∈ {ai , ai + 1, ai − 1, −ai } for 2 ≤ i ≤ n. 1, 1, −1 1, 1, 1 1, −1, 1 1, −1, −1 1, 2, 1 103. Sequences a1 , a2 , .

29. Dyck paths of length 2n + 2 whose first down step is followed by another down step. 30. Dyck paths of length 2n + 2 with no peak at height 1 and having leftmost peak at height 2 or 3. 22 Catalan Numbers 31. Dyck paths of length 2n + 2 for which the terminal descent is of even length and all other descents (if any) to the x-axis are of odd length, where a descent is a maximal sequence of consecutive down steps (compare item 26). 32. Dyck paths of length 4n such that every descent (maximal sequence of consecutive down steps) has length 2.