By Richard P. Stanley
Catalan numbers are essentially the most ubiquitous series of numbers in arithmetic. This e-book presents, for the 1st time, a finished selection of their houses and purposes in combinatorics, algebra, research, quantity thought, chance idea, geometry, topology, and different parts. After an advent to the elemental homes of Catalan numbers, the ebook provides 214 other forms of gadgets that are counted utilizing Catalan numbers, together with of workouts with strategies. The reader can attempt fixing the routines or just flick through them. sixty eight extra routines with prescribed hassle degrees current a number of houses of Catalan numbers and comparable numbers, equivalent to Fuss-Catalan numbers, Motzkin numbers, Schröder numbers, Narayana numbers, large Catalan numbers, q-Catalan numbers and (q,t)-Catalan numbers. The e-book concludes with a background of Catalan numbers through Igor Pak and a word list of key phrases. no matter if your curiosity in arithmetic is activity or learn, you will discover lots of attention-grabbing and stimulating evidence right here.
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Example text
Maximal chains ∅ = S0 ⊂ S1 ⊂ · · · ⊂ Sn = [n] of subsets of [n] such that Si − Si−1 = {m} if and only if m belongs to the rightmost maximal set of consecutive integers contained in Si . ∅ ⊂ 1 ⊂ 12 ⊂ 123, ∅ ⊂ 2 ⊂ 12 ⊂ 123, ∅ ⊂ 1 ⊂ 13 ⊂ 123 ∅ ⊂ 2 ⊂ 23 ⊂ 123, ∅ ⊂ 3 ⊂ 23 ⊂ 123 203. 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.
Partitions of [n] such that if a, e appear in a block B and b, d appear in a different block B where a < b < d < e, then there is a c ∈ B satisfying b < c < d. ) 165. , partitions of [n] such that if they are written with increasing entries in each block and blocks arranged in increasing order of their first entry, then the permutation of [n] obtained by erasing the dividers between the blocks is 231-avoiding. ) 166. Equivalence classes of the equivalence relation on the set Sn = {(a1 , . . , an ) ai = n} generated by (α, 0, β) ∼ (β, 0, α) if β (which may be ∈ Nn : empty) contains no 0’s.
123 213 132 312 231 116. Permutations a1 a2 · · · an of [n] for which there does not exist i < j < k and aj < ak < ai (called 312-avoiding permutations). 123 132 213 231 321 117. Permutations w of [2n] with n cycles of length two, such that the product (1, 2, . . , 2n) · w has n + 1 cycles (1, 2, 3, 4, 5, 6)(1, 2)(3, 4)(5, 6) = (1)(2, 4, 6)(3)(5) (1, 2, 3, 4, 5, 6)(1, 2)(3, 6)(4, 5) = (1)(2, 6)(3, 5)(4) (1, 2, 3, 4, 5, 6)(1, 4)(2, 3)(5, 6) = (1, 3)(2)(4, 6)(5) (1, 2, 3, 4, 5, 6)(1, 6)(2, 3)(4, 5) = (1, 3, 5)(2)(4)(6) (1, 2, 3, 4, 5, 6)(1, 6)(2, 5)(3, 4) = (1, 5)(2, 4)(3)(6) 118.