Exercises on Catalan and Related Numb ers excerpted from Enumerative Combinatorics,vol. 2 published by Cambridge University Press 1999 by Richard P. Stanley version of 23 June 1998 2n 1 19. [1]{[3+] Show that the Catalan numb ers C = count the number of elements n n+1 n of the 66 sets S , a i nnn given b elow. We illustrate the elements of each S for i i n = 3, hoping that these illustrations will makeany unde ned terminology clear. The terms used in vv{yy are de ned in Chapter 7. Ideally S and S should b e proved i j to have the same cardinality by exhibiting a simple, elegant bijection : S ! S ij i j so 4290 bijections in all. In some cases the sets S and S will actually coincide, but i j their descriptions will di er. a triangulations of a convex n + 2-gon into n triangles by n 1 diagonals that do not intersect in their interiors b binary parenthesizations of a string of n +1 letters xx xx xxx x x xxx xx xx xx xx c binary trees with n vertices r r r r r @ @ @ @ r r r r r @r @ @ @ @ r r r r d plane binary trees with 2n +1 vertices or n +1 endp oints r r r r r H @ @ H @ @ @ H @ r @r r r r r r r r r @ @ @ @ @ @ @ @ @ r @r r r r r r r r r r r @ @ @ @ @ @ r @r r @r r r r r e plane trees with n +1 vertices r r r r r @ @ @ r r r r r r r r r @ @ @ @ r r r r r @ r f planted i.e., ro ot has degree one trivalent plane trees with 2n +2 vertices 221 r r r r r r r r r r H @ @ @ @ H @ @ @ r r r r r r r r r r H @ @ @ @ @ @ @ @ r r r r r r r r r @r r @r @ @ @ @ @ @ r r r @r r r r @r g plane trees with n +2 vertices such that the rightmost path of each subtree of the ro ot has even length r r r r r ,l @ L @ @ r, r Lrlr r @r r r r r @ @ r r r r r @ @ r r r r r h lattice paths from 0; 0 to n; n with steps 0; 1 or 1; 0, never rising ab ove the line y = x r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r i Dyck paths from 0; 0 to 2n; 0, i.e., lattice paths with steps 1; 1 and 1; 1, never falling b elow the x-axis r @ r r r r r r @ @ @ @ @ r r r @r r r r r r r r r r r @ @ @ @ @ @ @ @ @ r @r r @r r @r @r r @r @r r @r @r @r j Dyck paths as de ned in i from 0; 0 to 2n +2;0 such that any maximal sequence of consecutive steps 1; 1 ending on the x-axis has odd length r r @ @ r r r r r @ @ @ r r r r r r r r @r r @ @ @ @ @ @ @ @ r @r @r @r @r r @r @r r r r r r @ @ @ @ r r r r r @ @ r r r r r @ @ @ @ @ @ r r r r r k Dyck paths as de ned in i from 0; 0 to 2n +2;0 with no p eaks at height two. 222 r r @ @ r r r r @ @ r r r r r r r r r r @ @ @ @ @ @ @ @ r @r @r @r @r r @r @r r @r @r r @ r r r r @ @ @ @ r r r r r @ @ r r r r @ @ r @r r @r l unordered pairs of lattice paths with n +1 steps each, starting at 0; 0, us- ing steps 1; 0 or 0; 1, ending at the same p oint, and only intersecting at the b eginning and end r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r m unordered pairs of lattice paths with n 1 steps each, starting at 0; 0, using steps 1; 0 or 0; 1, ending at the same p oint, such that one path never arises ab ove the other path n n nonintersecting chords joining 2n p oints on the circumference of a circle r r r r r r r r r r @ A @ r r r r r r r @r r r @ A @ @ r r @r r @r Ar r r r r o ways of connecting 2n p oints in the plane lying on a horizontal line by n nonin- tersecting arcs, each arc connecting two of the p oints and lying ab ove the p oints r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r p ways of drawing in the plane n +1 p oints lying on a horizontal line L and n arcs connecting them such that the arcs do not pass b elow L, the graph thus formed is a tree, notwo arcs intersect in their interiors i.e., the arcs are noncrossing, and atevery vertex, all the arcs exit in the same direction left or right A A A A A @ A s s @s As s s s As s s s @As s s s @As s s s s q ways of drawing in the plane n + 1 p oints lying on a horizontal line L and n arcs connecting them such that the arcs do not pass b elow L, the graph thus 223 formed is a tree, no arc including its endp oints lies strictly b elow another arc, and atevery vertex, all the arcs exit in the same direction left or right A A A A H H s s s s @ A s sHs @As s s s As s sHs @s s s s @As r sequences of n 1's and n 1's such that every partial sum is nonnegative with 1 denoted simply as b elow 111 111 111 111 111 s sequences 1 a a of integers with a i 1 n i 111 112 113 122 123 t sequences a <a < <a of integers satisfying 1 a 2i 1 2 n1 i 12 13 14 23 24 u sequences a ;a ;:::;a of integers such that a = 0 and 0 a a +1 1 2 n 1 i+1 i 000 001 010 011 012 v sequences a ;a ;:::;a of integers such that a 1 and all partial sums are 1 2 n1 i nonnegative 0; 0 0; 1 1; 1 1; 0 1; 1 w sequences a ;a ;:::;a of integers such that a 1, all partial sums are non- 1 2 n i negative, and a + a + + a =0 1 2 n 0;0;0 0;1;1 1;0;1 1;1;0 2;1;1 x sequences a ;a ;:::;a of integers such that 0 a n i, and such that if i<j, 1 2 n i a >0, a > 0, and a = a = = a = 0, then j i>a a i j i+1 i+2 j1 i j 000 010 100 200 110 y sequences a ;a ;:::;a of integers such that i a n and such that if i j a , 1 2 n i i then a a j i 123 133 223 323 333 z sequences a ;a ;:::;a of integers such that 1 a i and such that if a = j , 1 2 n i i then a j r for 1 r j 1 ir 111 112 113 121 123 224 aa equivalence classes B of words in the alphab et [n 1] such that any three con- secutive letters of any word in B are distinct, under the equivalence relation uij v uj iv for any words u; v and any i; j 2 [n 1] satisfying ji j j 2 f;g f1g f2g f12g f21g For n =4 a representative of each class is given by ;, 1, 2, 3, 12, 21, 13, 23, 32, 123, 132, 213, 321, 2132. bb partitions = ;:::; with n 1 so the diagram of is contained in 1 n1 1 0 0 0 an n 1 n 1 square, such that if = ; ;::: denotes the conjugate 1 2 0 partition to then whenever i i i i 0; 0 1; 0 1; 1 2; 1 2; 2 2 2 2 cc p ermutations a a a of the multiset f1 ; 2 ;:::;n g such that: i the rst o c- 1 2 2n currences of 1; 2;:::;n app ear in increasing order, and ii there is no subsequence of the form 112233 112332 122331 123321 122133 dd p ermutations a a a of the set [2n] such that: i 1; 3;:::;2n 1 app ear in 1 2 2n increasing order, ii 2; 4;:::;2n app ear in increasing order, and iii 2i 1 app ears b efore 2i, 1 i n 123456 123546 132456 132546 135246 ee p ermutations a a a of [n] with longest decreasing subsequence of length at 1 2 n most two i.e., there do es not exist i<j <k, a >a >a , called 321-avoiding i j k p ermutations 123 213 132 312 231 p ermutations a a a of [n] for which there do es not exist i < j < k and 1 2 n a <a <a called 312-avoiding p ermutations j k i 123 132 213 231 321 gg p ermutations w of [2n] with n cycles of length two, such that the pro duct 1; 2;:::;2n w has n + 1 cycles 1; 2; 3; 4; 5; 61; 23; 45; 6 = 12; 4; 635 1; 2; 3; 4; 5; 61; 23; 64; 5 = 12; 63; 54 1; 2; 3; 4; 5; 61; 42; 35; 6 = 1; 324; 65 1; 2; 3; 4; 5; 61; 62; 34; 5 = 1; 3; 5246 1; 2; 3; 4; 5; 61; 62; 53; 4 = 1; 52; 436 225 hh pairs u; v of p ermutations of [n] such that u and v have a total of n +1 cycles, and uv =1;2;:::;n 123 1; 2; 3 1; 2; 3 123 1; 23 1; 32 1; 32 12; 3 12; 3 1; 23 ii p ermutations a a a of [n] that can b e put in increasing order on a single stack, 1 2 n de ned recursively as follows: If ; is the empty sequence, then let S ; = ;.