By Richard P. Stanley

ISBN-10: 1107075092

ISBN-13: 9781107075092

Catalan numbers are the most ubiquitous series of numbers in arithmetic. This ebook presents, for the 1st time, a entire choice of their homes and purposes in combinatorics, algebra, research, quantity concept, chance thought, geometry, topology, and different components. After an advent to the elemental homes of Catalan numbers, the publication provides 214 other kinds of gadgets that are counted utilizing Catalan numbers, together with of routines with ideas. The reader can attempt fixing the routines or just flick through them. sixty eight extra routines with prescribed trouble degrees current a variety of houses of Catalan numbers and comparable numbers, comparable 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 means of Igor Pak and a thesaurus of key words. even if your curiosity in arithmetic is activity or study, you will discover lots of interesting and stimulating proof right here.

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Arrays a1 b1 a2 b2 · · · ar−1 · · · br−1 ar of integers, for some r ≥ 1, such that ai > 0, bi ≥ 0, bi < ai + bi−1 for 1 ≤ i ≤ r − 1 (setting b0 = 0). 1 0 1 1 0 2 1 0 2 1 1 1 0 2 ai = n, and 3 46 Catalan Numbers 175. Plane partitions with largest part at most two and contained in a rectangle of perimeter 2(n − 1) (including degenerate 0 × (n − 1) and (n − 1) × 0 rectangles) 0 1 2 176. Convex subsets S of the poset Z × Z, up to translation by a diagonal vector (m, m), such that if (i, j) ∈ S then 0 < i − j < n.

Ways to write (1, 1, . . , 1, −n) ∈ Zn+1 as a sum of vectors ei − ei+1 and ej − en+1 , without regard to order, where ek is the kth unit coordinate vector in Zn+1 . (1, −1, 0, 0) + 2(0, 1, −1, 0) + 3(0, 0, 1, −1) (1, 0, 0, −1) + (0, 1, −1, 0) + 2(0, 0, 1, −1) (1, −1, 0, 0) + (0, 1, −1, 0) + (0, 1, 0, −1) + 2(0, 0, 1, −1) (1, −1, 0, 0) + 2(0, 1, 0, −1) + (0, 0, 1, −1) (1, 0, 0, −1) + (0, 1, 0, −1) + (0, 0, 1, −1) 204. 5]) of width (number of columns) n + 1 such that each row begins strictly to the right of the beginning of the previous row and ends strictly to the right of the end of the previous row.

123 132 213 231 312 Bijective Exercises 39 125. Permutations w of [n] with the following property: Suppose that w has inversions, and let R(w) = {(a1 , . . , a ) ∈ [n − 1] : w = sa1 sa2 · · · sa }, where sj is as in item 124. Then a1 a2 · · · a = !. ,a )∈R(w) R(123) = {∅}, R(213) = {(1)}, R(312) = {(2, 1)}, R(231) = {(1, 2)} R(321) = {(1, 2, 1), (2, 1, 2)} 126. 128]) w-avoiding permutations of [n + 1], where w is any of 321, 312, or 231. w = 321 : 4123 3142 2413 3412 2341 w = 312 : 4321 3421 2431 2341 3241 w = 231 : 4123 4132 4213 4312 4321 127.

### Catalan Numbers by Richard P. Stanley

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