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 x x x xx x x xx x x x 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 11 1 11 1 1 11 1 1 1
s sequences 1 a a of integers with a i
1 n i
111 112 113 122 123
t sequences a 1 2 n 1 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 n 1 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 1 2 n i a >0, a > 0, and a = a = = a = 0, then j i>a a i j i+1 i+2 j 1 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 i r 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 n 1 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 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 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; 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 6 225 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 1 2 3 1; 2; 3 1; 2; 3 1 2 3 1; 2 3 1; 3 2 1; 3 2 1 2; 3 1 2; 3 1; 2 3 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 ; = ;. If w = unv is a sequence of distinct integers with largest term n, then S w = S u S v n. A stack-sortable p ermutation w is one for which S w = w. a a a 1 2 n ? For example, 4123 1 3 12 3 123 1234 2 - - - - 4 4 4 123 132 213 312 321 jj p ermutations a a a of [n] that can b e put in increasing order on two parallel 1 2 n queues. Now the picture is ... aa1 n 123 132 213 231 312 kk xed-p oint free involutions w of [2n] such that if i < j < k < l and w i = k , then w j 6= l in other words, 3412-avoiding xed-p oint free involutions 12 34 56 14 23 56 12 36 45 16 23 45 16 25 34 ll cycles of length 2n +1 in S with descent set fng 2n+1 2371456 2571346 3471256 3671245 5671234 mm Baxter p ermutations as de ned in Exercise 55 of [2n] or of [2n +1] that are reverse alternating as de ned at the end of Section 3.16 and whose inverses are reverse alternating 132546 153426 354612 561324 563412 1325476 1327564 1534276 1735462 1756342 226 nn p ermutations w of [n] such that if w has ` inversions then for all pairs of sequences ` a ;a ;:::;a ; b ;b ;:::;b 2 [n 1] satisfying 1 2 ` 1 2 ` w = s s s = s s s ; a a a b b b 1 2 1 2 ` ` where s is the adjacent transp osition j; j + 1 , we have that the `-element mul- j tisets fa ;a ;:::;a g and fb ;b ;:::;b g are equal thus, for example, w = 321 is 1 2 ` 1 2 ` not counted, since w = s s s = s s s , and the multisets f1; 2; 1g and f2; 1; 2g 1 2 1 2 1 2 are not equal 123 132 213 231 312 o o p ermutations w of [n] with the following prop erty: Supp ose that w has ` inver- sions, and let ` R w = f a ;:::;a 2 [n 1] : w = s s s g; 1 ` a a a 1 2 ` where s is as in nn . Then j X a a a = `!: 1 2 ` a ;:::;a 2R w 1 ` R 123 = f;g; R 213 = f 1 g; R 231 = f 1; 2 g R 312 = f 2; 1 g; R 321 = f 1; 2; 1 ; 2; 1; 2 g pp noncrossing partitions of [n], i.e., partitions = fB ;:::;B g 2 such that if 1 k n a i j 123 12 3 13 2 23 1 1 2 3 qq partitions fB ;:::;B g of [n] such that if the numb ers 1; 2;:::;n are arranged in 1 k order around a circle, then the convex hulls of the blo cks B ;:::;B are pairwise 1 k disjoint r r r r r T T T T r r r r r r r r r r T T rr noncrossing Murasaki diagrams with n vertical lines ss noncrossing partitions of some set [k ] with n + 1 blo cks, such that anytwo elements of the same blo ck di er by at least three 1 2 3 4 14 2 3 5 15 2 3 4 25 1 3 4 16 25 3 4 tt noncrossing partitions of [2n +1] into n +1 blo cks, such that no blo ck contains two consecutive integers 137 46 2 5 1357 2 4 6 157 24 3 6 17 246 3 5 17 26 35 4 227 uu nonnesting partitions of [n], i.e., partitions of [n] such that if a; e app ear in a 0 blo ck B and b; d app ear in a di erent blo ck B where a is a c 2 B satisfying b 123 12 3 13 2 23 1 1 2 3 The unique partition of [4] that isn't nonnesting is 14 23. vv Young diagrams that t in the shap e n 1;n 2;:::;1 ; ww standard Young tableaux of shap e n; n or equivalently, of shap e n; n 1 123 124 125 134 135 456 356 346 256 246 or 123 124 125 134 135 45 35 34 25 24 xx pairs P; Q of standard Young tableaux of the same shap e, each with n squares and at most two rows 12 12 12 13 13 12 13 13 123; 123 3 ; 3 3 ; 2 2 ; 3 2 ; 2 yy column-strict plane partitions of shap e n 1;n 2;:::;1 , such that eachentry in the ith row is equal to n i or n i +1 33 33 32 32 22 2 1 2 1 1 zz convex subsets S of the p oset Z Z, up to translation by a diagonal vector m; m , such that if i; j 2 S then 0 ; f 1; 0 g f 2; 0 g f 1; 0 ; 2; 0 g f 2; 0 ; 2; 1 g aaa linear extensions of the p oset 2 n r 6 @ 123456 @ r r @ 123546 4 5 @ 132456 @ r r @ 2 3 132546 @ @ 135246 r @ 1 228 r 33 r 23 @ @ r @ 32 @ @ @r 13 @ @ r 22 @ @ r 31 r 12 r 21 r 11 Figure 5: A p oset with C = 14 order ideals 4 bbb order ideals of Int n 1 , the p oset of intervals of the chain n 1 r r c @ ;; a; b; ab; abc @ r r @r a b 2 Int 2 ccc order ideals of the p oset A obtained from the p oset n 1 n 1 by adding n the relations i; j < j; i if i>j see Figure 5 for the Hasse diagram of A 4 ; f11g f11; 21g f11; 21; 12g f11; 21; 12; 22g ddd nonisomorphic n-element p osets with no induced subp oset isomorphic to 2 + 2 or 3 + 1 r r r r r r r r r r @ @ @ @ r r r r r eee nonisomorphic n + 1 -element p osets that are a union of twochains and that are not a nontrivial ordinal sum, ro oted at a minimal element r r r r r r r r r r r r f @ @ r rf r f@r r @rf r rf f relations R on [n] that are re exive iR i , symmetric iR j jRi , and such that if 1 i ij for the pair i; j , and we omit the pairs ii ; f12; 21g f23; 32g f12; 21; 23; 32g f12; 21; 13; 31; 23; 32g ggg joining some of the vertices of a convex n 1 -gon by disjoint line segments, and circling a subset of the remaining vertices r r rf r rf r r r rf rf hhh ways to stack coins in the plane, the b ottom row consisting of n consecutive coins m m m mm mm mmm mmm mmm mm m mm m iii n-tuples a ;a ;:::;a ofintegers a 2 such that in the sequence 1a a a 1, 1 2 n i 1 2 n each a divides the sum of its two neighb ors i 14321 13521 13231 12531 12341 jjj n-element multisets on Z= n +1 Z whose elements sum to 0 000 013 022 112 233 kkk n-element subsets S of N N such that if i; j 2 S then i j and there is a lattice path from 0; 0 to i; j with steps 0; 1 , 1; 0 , and 1; 1 that lies entirely inside S f 0; 0 ; 1; 0 ; 2; 0 g f 0; 0 ; 1; 0 ; 1; 1 g f 0; 0 ; 1; 0 ; 2; 1 g f 0; 0 ; 1; 1 ; 2; 1 g f 0; 0 ; 1; 1 ; 2; 2 g n lll regions into which the cone x x x in R is divided by the hyp erplanes 1 2 n x x =1, for 1 i i j intersected with the hyp erplane x + x + x =0 1 2 3 230 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3 2 1 5 1 2 3 1 3 2 1 5 2 5 1 4 4 1 5 2 2 5 1 4 3 2 3 3 3 2 3 3 3 2 3 1 5 2 2 5 1 4 4 1 5 2 3 1 3 2 1 5 1 2 1 1 1 1 1 1 1 1 Figure 6: The frieze pattern corresp onding to the sequence 1; 3; 2; 1; 5; 1; 2; 3 mmm p ositive integer sequences a ;a ;:::;a for which there exists an integer array 1 2 n+2 necessarily with n +1 rows 1 1 1 1 1 1 1 1 a a a a a a a 1 2 3 n+2 1 2 n 1 b b b b b b 1 2 3 n+2 1 n 2 r r r r r 1 2 3 n+2 1 1 1 1 1 54 r st such that any four neighb oring entries in the con guration satisfy st = ru +1 u an example of such an array for a ;:::;a = 1; 3; 2; 1; 5; 1; 2; 3 necessarily 1 8 unique is given by Figure 6 : 12213 22131 21312 13122 31221 nnn n-tuples a ;:::a of p ositiveintegers such that the tridiagonal matrix 1 n 3 2 a 1 0 0 0 0 1 7 6 1 a 1 0 0 0 2 7 6 7 6 0 1 a 1 0 0 3 7 6 7 6 7 6 7 6 7 6 7 6 7 6 5 4 0 0 0 0 a 1 n 1 0 0 0 0 1 a n is p ositive de nite with determinant one 131 122 221 213 312 20. a [2+] Let m; n be integers satisfying 1 n the number of lattice paths from 1; 0 to m; n with steps 0; 1 and 1; 0 that intersect the line y = x in at least one p oint is equal to the number of lattice paths from 0; 1 to m; n with steps 0; 1 and 1; 0 . 231 b [2{] Deduce that the numb er of lattice paths from 0; 0 to m; n with steps 1; 0 m+n m n and 0; 1 that intersect the line y = x only at 0; 0 is given by . m+n n c [1+] Show from b that the number of lattice paths from 0; 0 to n; n with steps 1; 0 and 0; 1 that never rise ab ove the line y = x is given by the Catalan 2n 1 number C = . This gives a direct combinatorial pro of of interpretation n n+1 n h of C in Exercise 19. n 2n 21. a [2+] Let X b e the set of all lattice paths from 0; 0 to n; n with steps 0; 1 n n and 1; 0 . De ne the excedance also sp elled \exceedance" of a path P 2 X n 0 to be the number of i such that at least one p oint i; i of P lies ab ove the line 0 y = x i.e., i > i . Show that the number of paths in X with excedance j is n indep endent of j . b [1] Deduce that the number of P 2 X that never rise ab ove the line y = x is n 2n 1 given by the Catalan number C = a direct pro of of interpretation h n n+1 n of C in Exercise 19 . Compare with Example 5.3.11, which also gives a direct n 2n 1 as well as combinatorial interpretation of C when written in the form n n+1 n 2n+1 1 in the form . 2n+1 n 22. [2+] Show bijectively if p ossible that the number of lattice paths from 0; 0 to 2n; 2n with steps 1; 0 and 0; 1 that avoid the p oints 2i 1; 2i 1 , 1 i n,is equal to the Catalan number C . 2n 23. [3{] Consider the following chess p osition. Z Z ZkZ o Z Z Z pZ ZPO Z Z Z Z Z Z Z Z Z Z Z Z ZK A Z Z Z Z Z Z Z Black is to make 19 consecutive moves, after which White checkmates Black in one move. Black may not move into check, and may not check White except p ossibly on his last move . Black and White are cooperating to achieve the aim of checkmate. In chess problem parlance, this problem is called a serieshelpmate in 19. How many di erent solutions are there? 232 24. [?] Explain the signi cance of the following sequence: un, dos, tres, quatre, cinc, sis, set, vuit, nou, deu, ::: 2n 1 25. [2]{[5] Show that the Catalan number C = has the algebraic interpretations n n+1 n given b elow. a number of two-sided ideals of the algebra of all n 1 n 1 upp er triangular matrices over a eld b dimension of the space of invariants of SL 2; C acting on the 2nth tensor power 2n T V ofits \de ning" two-dimensional representation V c dimension of the irreducible representation of the symplectic group Sp 2 n 1 ; C or Lie algebra sp 2 n 1 ; C with highest weight , the n 1 st fundamental n 1 weight d dimension of the primitive intersection homology say with real co ecients of the toric variety asso ciated with a rationally emb edded n-dimensional cub e e the generic number of PGL 2; C equivalence classes of degree n rational maps with a xed branch set f numb er of translation conjugacy classes of degree n + 1 monic p olynomials in one complex variable, all of whose critical p oints are xed g dimension of the algebra over a eld K with generators ;:::; and relations 1 n 1 2 = i i = ; if ji j j =1 i j i i = ; if ji j j 2; i j j i where is a nonzero elementofK + h number of -sign typ es indexed by A the set of p ositive ro ots of the ro ot n 1 system A n 1 i Let the symmetric group S act on the p olynomial ring A = C [x ;:::;x ;y ;:::;y ] n 1 n 1 n by wf x ;:::;x ;y ;:::;y =f x ;:::;x ;y ;:::;y for all w 2 S . 1 n 1 n n w 1 w n w 1 w n Let I b e the ideal generated by all invariants of p ositive degree, i.e., I = hf 2 A : w f = f for all w 2 S ; and f 0 =0i: n Then conjecturally C is the dimension of the subspace of A=I a ording the n sign representation, i.e., C = dimff 2 A=I : w f = sgn w f for all f 2 S g: n n 26. a [3{] Let D be aYoung diagram of a partition , as de ned in Section 1.3. Given a square s of D let t be the lowest square in the same column as s, and let u be the rightmost square in the same rowass. Let f s b e the numb er of paths from 233 t to u that stay within D , and such that each step is one square to the north or one square to the east. Insert the number f s in square s, obtaining an array A. For instance, if = 5;4;3;3 then A is given by 16 7 2 1 1 6 3 1 1 3 2 1 1 1 1 Let M be the largest square subarray using consecutive rows and columns of A containing the upp er left-hand corner. Regard M as a matrix. For the ab ove example we have 2 3 16 7 2 4 5 6 3 1 : M = 3 2 1 Show that det M =1. b [2] Find the unique sequence a ;a ;::: of real numb ers such that for all n 0we 0 1 have 3 2 3 2 a a a a a a 1 2 n 0 1 n 7 6 7 6 a a a a a a 2 3 n+1 1 2 n+1 7 6 7 6 7 6 7 6 7 6 7 6 =1: = det det 7 6 7 6 7 6 7 6 5 4 5 4 a a a a a a n n+1 2n 1 n n+1 2n When n = 0 the second matrix is empty and by convention has determinant one. 27. a [3{] Let V be a real vector space with basis x ;x ;:::;x and scalar pro duct n 0 1 n de ned by hx ;x i = C , the i + j -th Catalan numb er. It follows from Ex- i j i+j ercise 26 b that this scalar pro duct is p ositive de nite, and therefore V has an orthonormal basis. Is there an orthonormal basis for V whose elements are inte- n gral linear combinations of the x 's ? i b [3{] Same as a , except now hx ;x i = C . i j i+j+1 c [5{] Investigate the same question for the matrices M of Exercise 26 a so hx ;x i = M when is self-conjugate so M is symmetric . i j ij 28. a [3{] Supp ose that real numb ers x ;x ;:::;x are chosen uniformly and inde- 1 2 d p endently from the interval [0; 1]. Show that the probability that the sequence 1 2 x ;x ;:::;x is convex i.e., x x + x for 2 i d 1 is C = d 1 ! , 1 2 d i i 1 i+1 d 1 2 where C denotes a Catalan numb er. d 1 234 d b [3{] Let C denote the set of all p oints x ;x ;:::;x 2 R such that 0 x 1 d 1 2 d i and the sequence x ;x ;:::;x is convex. It is easy to see that C is a d- 1 2 d d dimensional convex p olytop e, called the convexotope. Show that the vertices of C consist of the p oints d j 1 j 2 1 1 2 1; ; ;:::; ;0;0;:::;0; ; ;:::;1 55 j j j k k d+1 with at least one 0 co ordinate , together with 1; 1;:::;1 so +1 vertices 2 1 ; 1 , 1; 0; 0 , in all . For instance, the vertices of C are 0; 0; 0 , 0; 0; 1 , 0; 3 2 1 1; ; 0 , 1; 0; 1 , 1; 1; 1 . 2 c [3] Show that the Ehrhart quasi-p olynomial i C ;n of C as de ned in Section d d 4.6 is given by X n y := i C ;n x d d n0 ! d d 1 X X 1 1 1 1 1 ; 56 = 1 x [1][r 1]! [1][d r ]! [1][r 1]! [1][d 1 r ]! r=1 r =1 i where [i] = 1 x , [i]! = [1][2] [i], and denotes Hadamard pro duct. For instance, 1 y = 1 2 1 x 1+x y = 2 3 1 x 2 1+2x+3x y = 3 3 2 1 x 1 x 2 3 4 5 6 1+3x +9x +12x +11x +3x +x y = 4 2 2 2 3 1 x 1 x 1 x 2 3 4 5 6 7 8 9 10 1+4x +14x +34x +63x +80x +87x +68x +42x +20x +7x y = : 5 2 2 3 2 4 1 x 1 x 1 x 1 x Is there a simpler formula than 56 for i C ;n ory ? d d 29. [3] Supp ose that n +1 p oints are chosen uniformly and indep endently from inside a square. Show that the probability that the p oints are in convex p osition i.e., each 2 p ointisavertex of the convex hull of all the p oints is C =n! . n 30. [3{] Let f be the number of partial orderings of the set [n] that contain no induced n subp osets isomorphic to 3 + 1 or 2 + 2. This exercise is the lab elled analogue of Exercise 19 ddd . As mentioned in the solution to this exercise, such p osets are called 2 3 semiorders. Let C x =1+x+2x +5x + b e the generating function for Catalan numb ers. Show that n X x x = C 1 e ; 57 f n n! n0 1 1 x 2 3 the comp osition of C x with the series 1 e = x x + x . 2 6 235 2 r a a a J J J 2 r a a a J 2 2 J r a a 2 r a a a @ @ 2 @r a a a Figure 7: The Tamari lattice T 3 d+1 31. a [3{] Let P denote the convex hull in R of the origin together with all vectors e e , where e is the ith unit co ordinate vector and i < j . Thus P is a d- i j i dimensional convex p olytop e. Show that the relative volume of P as de ned in Section 4.6 is equal to C =d!, where C denotes a Catalan numb er. d d b [3] Let i P ;n denote the Ehrhart p olynomial of P . Find a combinatorial inter- pretation of the co ecients of the i-Eulerian p olynomial in the terminology of Section 4.3 X n d+1 i P ;n x : 1 x n0 32. a [3{] De ne a partial order T on the set of all binary bracketings parenthesiza- n tions of a string of length n + 1 as follows. Wesay that v covers u if u contains a sub expression xy z where x, y , z are bracketed strings and v is obtained from 2 2 2 2 u by replacing xy z with x yz . For instance, a a a a a is covered by 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 a a a a a , a a a a a , a a a a a a , and a a a a a . Figures 7 and 8 show the Hasse diagrams of T and T . In Figure 8, we have 3 4 enco ded the binary bracketing by a string of four +'s and four 's, where a + stands for a left parenthesis and a for the letter a, with the last a omitted. Let U b e the p oset of all integer vectors a ;a ;:::;a such that i a n and n 1 2 n i such that if i j a then a a , ordered co ordinatewise. Show that T and i j i n U are isomorphic p osets. n b [2] Deduce from a that T is a lattice called the Tamari lattice . n 33. Let C be a convex n-gon. Let S be the set of all sets of diagonals of C that do not intersect in the interior of C . Partially order the elementofS by inclusion, and add a ^ 1. Call the resulting p oset A . n a [3{] Show that A is a simplicial Eulerian lattice of rank n 2, as de ned in n Section 3.14. b [3] Show in fact that A is the lattice of faces of an n 3 -dimensional convex n p olytop e Q . n c [3{] Find the number W = W n of elements of A of rank i. Equivalently, W is i i n i the number of ways to draw i diagonals of C that do not intersect in their interiors. Note that by Prop osition 2.1, W n is also the number of plane trees with n + i i vertices and n 1 endp oints such that no vertex has exactly one successor. 236 +-+-+-+- +-+-++-- +-++--+- ++--+-+- +-++-+-- +-+++--- ++-+-+-- ++-+--+- ++--++-- ++-++--- +++-+--- +++--+-- +++---+- ++++---- Figure 8: The Tamari lattice T 4 237 d [3{] De ne n 3 n 3 X X n i 3 n 3 i W x 1 = h x ; 58 i i i=0 i=0 as in equation 3.44 . The vector h ;:::;h is called the h-vector of A or 0 n 3 n of the p olytop e Q . Find an explicit formula for each h . n i 34. There are many p ossible q -analogues of Catalan numb ers. In a we give what is p erhaps the most natural \combinatorial" q -analogue, while in b we give the most natural \explicit formula" q -analogue. In c we give an interesting extension of b , while d and e are concerned with another sp ecial case of c . a [2+] Let X A P C q = q ; n P where the sum is over all lattice paths P from 0; 0 to n; n with steps 1; 0 and 0; 1 , such that P never rises ab ove the line y = x, and where A P is the area under the path and ab ove the x-axis . Note that by Exercise 19 h , we have C 1 = C . It is interesting to see what statistic corresp onds to A P n n for many of the other combinatorial interpretations of C given in Exercise 19. n 2 3 For instance, C q = C q = 1, C q = 1+q, C q = 1+q +2q +q , 0 1 2 3 2 3 4 5 6 C q = 1+q+2q +3q +3q +3q +q . Show that 4 n X i+1 n i C q = C q C q q : n+1 i n i i=0 n ~ 2 C 1=q , then the generating function Deduce that if C q =q n n X n ~ F x = C q x n n0 satis es xF x F qx F x +1 = 0: From this we get the continued fraction expansion 1 : 59 F x = x 1 qx 1 2 q x 1 1 b [2+] De ne 2n 1 c q = : n n+1 n 238 2 2 3 4 6 For instance, c q = c q = 1, c q = 1+ q , c q = 1+q +q +q + q , 0 1 2 3 2 3 4 5 6 7 8 9 10 12 c q = 1+q +q +2q +q +2q +q +2q +q +q + q . Show that 4 X ma j w c q = q ; n w where w ranges over all sequences a a a of n 1's and n 1's such that each 1 2 2n partial sum is nonnegative, and where X ma j w = i; fi:a>a g i i+1 the ma jor index of w . c [3{] Let t be a parameter, and de ne n X n n 1 2 i +it q : c t; q = n n+1 i i +1 i=0 Show that X ma j w + t 1 des w c t; q = q ; n w where w ranges over the same set as in b , and where des w =fi : a >a g; i i+1 the number of descents of w . Hence c 1; q = c q . n n d [3{] Show that 1+q c 0; q = c q : n n n 1+q 2 3 4 For instance, c 0; q = c 0; q = 1, c 0; q =1+q, c 0; q = 1+q+q +q +q , 0 1 2 3 2 3 4 5 6 7 8 9 c 0; q =1+q+q +2q +2q +2q +2q +q +q +q . 4 P i e [3+] Show that the co ecients of c 0; q are unimodal, i.e., if c 0; q = bq , n n i then for some j we have b b b b b . In fact, we can 0 1 j j+1 j +2 1 1 2 take j = b deg c 0; q c = b n 1 c. n 2 2 35. Let Q b e the p oset of direct-sum decomp ositions of an n-dimensional vector space V n n ^ over the eld F , as de ned in Example 5.5.2 b . Let Q denote Q with a 0 adjoined, q n n ^ ^ 0;1 . Hence by 5.74 we have and let q = n Q n n n X X x x q = log : n n n 2 2 q n ! q n ! n1 n0 a [3{] Show that 1 n 2 n 1 q = 1 q 1 q 1 q 1 P q ; n n n 239 n where P q is a p olynomial in q of degree with nonnegative integral co e- n 2 2n 1 cients, satisfying P 1 = . For instance, n n P q = 1 1 P q = 2+q 2 2 3 P q = 3+3q +3q +q 3 2 3 2 3 P q = 2+2q +q 2+2q +2q +q : 4 b Show that n X X n n x n 2 2 = q C 1=q x ; exp q P 1=q n n n n1 n1 where C q isthe q-Catalan p olynomial de ned in Exercise 34 a . n 36. a [2+] The Narayana numbers N n; k are de ned by n n 1 : N n; k = n k k 1 Let X b e the set of all sequences w = w w w of n 1's and n 1's with all nk 1 2 2n partial sums nonnegative, such that k =fj :w =1;w = 1g: j j+1 Give a combinatorial pro of that N n; k =X . Hence by Exercise 19 r , there nk follows n X N n; k =C : n k =1 It is interesting to nd for each of the combinatorial interpretations of C given n by Exercise 19 a corresp onding decomp osition into subsets counted by Narayana numb ers. P P n k b [2+] Let F x; t = N n; k x t . Using the combinatorial interpreta- n1 k 1 tion of N n; k given in a , show that 2 xF + xt + x 1 F + xt =0; 60 so p 2 2 1 x xt 1 x xt 4x t F x; t = : 2x 37. [2+] The Motzkin numbers M are de ned by n p 2 X 1 x 1 2x 3x n M x = n 2 2x n0 2 3 4 5 6 7 8 = 1+x +2x +4x +9x +21x +51x + 127x + 323x 9 10 +835x + 2188x + : n 2n Show that M = C and C = M , where C denotes a Catalan numb er. n 1 n 0 n 240 38. [3{] Show that the Motzkin number M has the following combinatorial interpretations. n See Exercise 46 b for an additional interpretation. a Numberofways of drawing anynumb er of nonintersecting chords among n p oints on a circle. b Number of walks on N with n steps, with steps 1, 0, or 1, starting and ending at 0. c Number of lattice paths from 0; 0 to n; n , with steps 0; 2 , 2; 0 , and 1; 1 , never rising ab ove the line y = x. d Number of paths from 0; 0 to n; n with steps 1; 0 , 1; 1 , and 1; 1 , never going b elow the x-axis. Such paths are called Motzkin paths. e Number of pairs 1 a < < a n and 1 b < < b n of integer 1 k 1 k sequences such that a b and every integer in the set [n] app ears at least once i i among the a 's and b 's. i i f Number of ballot sequences as de ned in Corollary 2.3 ii a ;:::;a such 1 2n+2 that we never have a ;a ;a = 1; 1;1 . i 1 i i+1 g Numb er of plane trees with n=2 edges, allowing \half edges" that have no succes- sors and count as half an edge. h Number of plane trees with n +1 edges in which no vertex, the ro ot excepted, has exactly one successor. i Number of plane trees with n edges in which every vertex has at most two suc- cessors. j Numb er of binary trees with n 1 edges such that no two consecutive edges slant to the right. k Number of plane trees with n +1 vertices such that every vertex of odd height with the ro ot having height0 has at most one successor. l Number of noncrossing partitions = fB ;:::;B g of [n] as de ned in Exer- 1 k cise 3.68 such that if B = fbg and a < b < c, then a and c app ear in di erent i blo cks of . m Number of noncrossing partitions of [n +1] such that no blo ck of contains two consecutive integers. 39. [3{] The Schroder numb ers r and s were de ned in Section 2. Show that they have n n the following combinatorial interpretations. a s is the total numb er of bracketings parenthesizations of a string of n letters. n 1 b s is the number of plane trees with no vertex of degree one and with n end- n 1 p oints. c r is the numb er of plane trees with n vertices and with a subset of the endp oints n 1 circled. 241 d s is the numb er of binary trees with n vertices and with each right edge colored n either red or blue. e s is the numb er of lattice paths in the x; y plane from 0; 0 to the x-axis using n steps 1;k , where k 2 P or k = 1, never passing b elow the x-axis, and with n steps of the form 1; 1 . f s is the number of lattice paths in the x; y plane from 0; 0 to n; n using n steps k; 0 or 0; 1 with k 2 P, and never passing ab ove the line y = x. g r is the number of parallelogram p olynomino es de ned in the solution to n 1 Exercise 19 l of p erimeter 2n with each column colored either black or white. h s is the number of ways to drawanynumb er of diagonals of a convex n + 2 -gon n that do not intersect in their interiors i s is the number of sequences i i i , where i 2 P or i = 1 and k can be n 1 2 k j j arbitrary , such that n = fj : i = 1g, i + i + + i 0 for all j , and j 1 2 j i + i + + i =0. 1 2 k j r is the numb er of lattice paths from 0; 0 to n; n , with steps 1; 0 , 0; 1 , and n 1; 1 , that never rise ab ove the line y = x. k r is the number of n n p ermutation matrices P with the following prop erty: n 1 We can eventually reach the all 1's matrix by starting with P and continually replacing a 0 by a 1 if that 0 has at least two adjacent 1's, where an entry a is ij de ned to b e adjacent to a and a . i1;j i;j 1 l Let u = u u 2 S . We say that a p ermutation w = w w 2 S is u- 1 k k 1 n n