Download Catalan Numbers by Richard P. Stanley PDF

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.

Show description

Read Online or Download Catalan Numbers PDF

Best discrete mathematics books

Comprehensive Mathematics for Computer Scientists

  This two-volume textbook entire arithmetic for the operating machine Scientist is a self-contained complete presentation of arithmetic together with units, numbers, graphs, algebra, good judgment, grammars, machines, linear geometry, calculus, ODEs, and particular topics equivalent to neural networks, Fourier thought, wavelets, numerical matters, information, different types, and manifolds.

Algebraic Semantics of Imperative Programs

Algebraic Semantics of relevant courses provides a self-contained and novel "executable" advent to formal reasoning approximately critical courses. The authors' basic target is to enhance programming skill by way of enhancing instinct approximately what courses suggest and the way they run. The semantics of principal courses is laid out in a proper, applied notation, the language OBJ; this makes the semantics hugely rigorous but easy, and offers aid for the mechanical verification of application homes.

Structured Matrices in Mathematics, Computer Science, and Engineering II

Many very important difficulties in technologies, arithmetic, and engineering may be lowered to matrix difficulties. in addition, quite a few functions frequently introduce a distinct constitution into the corresponding matrices, in order that their entries should be defined by means of a definite compact formulation. vintage examples contain Toeplitz matrices, Hankel matrices, Vandermonde matrices, Cauchy matrices, decide matrices, Bezoutians, controllability and observability matrices, and others.

An Engineer’s Guide to Mathematica

An Engineers advisor to Mathematica allows the reader to realize the talents to create Mathematica nine courses that clear up a variety of engineering difficulties and that reveal the consequences with annotated pix. This e-book can be utilized to benefit Mathematica, as a better half to engineering texts, and in addition as a reference for acquiring numerical and symbolic options to quite a lot of engineering subject matters.

Additional info for Catalan Numbers

Sample text

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.

Download PDF sample

Rated 4.03 of 5 – based on 20 votes