Dyck paths with coloured ascents

0.22 MB·English

by  Andrei Asinowski

DYCK PATHS WITH COLOURED ASCENTS
7
0 ANDREIASINOWSKI† ANDTOUFIKMANSOUR‡
0
2

n
a †Caesarea Rothschild Institute, University of Haifa, Haifa 31905, Israel, [phone]
J ‡Department of Mathematics, University of Haifa, Haifa 31905, Israel, [phone]

5
2 [email protected], [email protected]

]
Abstract
O
C
We introduce a notion of Dyck paths with coloured ascents. For several ways of colouring, we
.
h establishbijectionsbetweensetsofsuchpathsandothercombinatorialstructures,suchasnon-crossing
t trees, dissections of a convex polygon, etc. In some cases enumeration gives new expression for
a
m sequences enumerating these structures.
[
Keywords. Dyck paths, non-crossing graphs, dissections of polygon by diagonals.

1 2000 Mathematics Subject Classification: 05A05, 05A15.
v
3
3 1. Introduction
7
1 1.1. Coloured Dyck paths. ADyck path of length 2nis asequenceP ofletters U andD,suchthat
0 #(U)=#(D)=n (where#means “numberof”)inP,and#(U) #(D) inanyinitialsubsequence
7 ≥
of P. A Dyck path of length 2n is usually represented graphically as a lattice path from the point
0
(0,0) to the point (2n,0) that does not pass below the x-axis, where U is the “up step” (1,1) and
/
h
D is the “down step” (1, 1). The set of all Dyck paths of length 2n will be denoted by (n). We
t − D
a shall also denote = . It is well known that (n) is equal to the n-th Catalan number
m C = 1 2n (seeD[10,SPna≥ge0D22n2, Exercise 19(i)]). |D |
n n+1 n
v: A maxi(cid:0)ma(cid:1)l subsequence of k consecutive U’s (that is, not preceded or followed by another U) in
i a Dyck path will be called a k-ascent and denoted by Uk. Similarly, a maximal subsequence of k
X
consecutiveD’s will be denotedby Dk. The Dyck pathUkDk willbe calledpyramid of length 2k and
r
a denoted by Λk.
In this paper we present a new generalization of Dyck paths. Let L = , , ,... be a
0 1 2
{L L L }
sequence of sets, and let a = . We colour all ascents in a Dyck path, when the set of colours
k k
|L |
for each k-ascent is . In this way we obtain Dyck paths with ascents coloured by L (shortly Dyck
k
L
paths coloured by L, or coloured Dyck paths). Each Dyck path P produces thus a coloured Dyck
i
paths, when the product is taken over the lengths of all ascents in P. ColouredQDyck paths will be
denotedbycapitalletterswith“hat”,e.g.Pˆ. ThepyramidΛk withUk colouredbyaspecifiedcolour
C (k) will be denoted by Λk C .
∈L h i
The set of all Dyck paths of length 2n coloured by members of L will be denoted by L(n). We
shallalsodenote L = L(n). In orderto obtainageneralexpressionenumerating DL(n), we
D n≥0D |D |
note that any Dyck patSh can be presented uniquely in the form
UkDP DP ...DP ,
k k−1 1
1
2 ANDREIASINOWSKI† ANDTOUFIKMANSOUR‡
whereP ,P ,...,P are(possiblyempty)Dyckpaths. ThereforeM(x),the generatingfunctionfor
k k−1 1
the sequence L(n) , satisfies
n≥0
{|D |}
M(x)=a +a xM(x)+a x2M2(x)+...,
0 1 2
or
(1) M =A(xM),
where A(x)= a xi is the generating function for the sequence .
i≥0 i {|Li|}i≥0
However,wePshall consider rather ’s than merely a ’s, and we shall establish bijections between
k k
L(n) for some specific L, and otheLr combinatorial structures. For example, in our main result
D
= (k), and thus a = C = 1 2k . We shall show the bijection between Dyck paths coloured
Lk D k k k+1 k
in this way and non-crossing trees to(cid:0)be(cid:1)defined in Section 1.2.
1.2. Non-crossing trees. A non-crossing tree (an “NC-tree”) on [n] is a labeled tree which can be
representedbyadrawinginwhichtheverticesarepointsonacircle,labeledby 1,2,...,n clockwise,
{ }
and the edges are non-crossing straight segments. The vertex 1 will be also called the root, and we
shall depict it as a top vertex. Non-crossing trees have been studied by Chen et al. [2], Deutsch et
al. [3, 4], Flajolet et al. [5], Hough [6], Noy et al. [7], and Panholzer et al. [9]. Denote the set of all
NC-trees on [n] by (n). It is well known that (n+1) = 1 3n .
NC |NC | 2n+1 n
(cid:0) (cid:1)
Weshallusethefollowingnotion. If a,b isanedgeofT,weshallratherdenoteitby(a,b),always
{ }
assuming that a < b. For a vertex v, we denote N−(v) = a V(T) : (a,v) E(T) (the in-edges
{ ∈ ∈ }
incident to v), N+(v) = b V(T) : (v,b) E(T) (the out-edges incident to v); d−(v) = N−(v)
{ ∈ ∈ } | |
(the in-degree of v), d+(v)= N+(v) (the out-degree of v).
| |
Consider the NC-trees on [n] with the property: For each vertex v = 1, we have d−(v) = 1. Such
6
trees will be called non-crossing out-trees(“NCO-trees”);the set ofNCO-trees on [n] will be denoted
by (n).
NCO
We also denote = (n+1) and = (n+1).
NC n≥0NC NCO n≥0NCO
S S
1.3. The results. We establish bijections between L(n) and other combinatorial structures, for a
D
few specific choices of L. The main result is the following theorem:
Theorem 1. There is a bijection between the set of Dyck paths of length 2n with k-ascents coloured
by Dyck paths of length 2k and the set of non-crossing trees on [n+1].
Otherresultsarespecialcasesandvariationsofthis theorem. Substituting the generationfunction
of Lin (1)enables us to enumerateeasily the combinatorialstructures being in bijection with L(n).
D
2. Dyck paths coloured by Dyck paths
Let D = (0), (1), (2),... . In this Section we consider D(n) – the set of Dyck paths of
{D D D } D
length 2n with k-ascents coloured by Dyck paths of length 2k, i. e. we take = (k).
k
L D
First we introduce a convenient way to depict thus coloured Dyck paths. Given a k-ascent Uk
coloured by a Dyck path C of length 2k, we draw a copy of C, rotated by 45◦ and scaled by 1/√2,
between the endpoints of Uk. Figure 1 presents in this way the Dyck path U5D2U3D6 with U5
coloured by UUDUUDDDUD and U3 coloured by UUDUDD.
DYCK PATHS WITH COLOURED ASCENTS 3
Figure 1. A Dyck path with k-ascents coloured by Dyck paths of length 2k.
2.1. Enumeration. Let us enumerate D(n). The generating function for (n) is
n≥0
D {|D |}
1 √1 4x
− − =1+x+2x2+5x3+....
2x
Substituting this in (1), we obtain
1 √1 4xM
M = − − .
2xM
After simplifications,we haveM 1=xM3. Denoting L=M 1andapplying Lagrange’sinversion
− −
formula (see [10, Section 5.4] and [11, Section 5.1]) on L = x(L+1)3, we get that the coefficient of
xn in L is
1 1 3n 1 3n
[xn]L= [Ln−1](L+1)3n = = .
n n(cid:18)n 1(cid:19) 2n+1(cid:18)n(cid:19)
−
Thus we have D(n) = (n+1).
|D | |NC |
In this Section will shall construct, for each n 0, a bijective function ϕ : D(n) (n+1).
n
It will be presented as a restriction of a bijective≥function ϕ : D . TDhe func→tionNϕC will be
D → NC
constructedbythefollowingsteps: InSubsection2.2wedescribearecursiveprocedureofdecomposing
aDyckpathintopyramids. InSubsection2.3weconstructabijectionϑ: . InSubsection2.4
D →NCO
we firstdefine ϕ forcolouredpyramidsandthen, using observationsfromSubsection2.2, for allDyck
paths coloured by D. All by all, this will give us the function ϕ, and we shall also show that it is
bijective.
2.2. Decomposition of a Dyck path into pyramids. Let P be a Dyck path. Recall that it can
be presented uniquely in the form
(2) P =UkDP DP ...DP ,
k k−1 1
where P ,P ,...,P are (possibly empty) Dyck paths. We say that Λk is the base pyramid of P
k k−1 1
and that P ,P ,...,P are appended to Λk, and denote this by P = Λk [P ,P ,...,P ]. This
k k−1 1 k k−1 1
∗
will be called a [primary] decomposition of P. If P = P = = P = , then P is a pyramid
k k−1 1
··· ∅
Λk, and we stop, identifying Λk [ , ,..., ] with Λk. Otherwise we decompose nonempty paths
∗ ∅ ∅ ∅
among P ,P ,...,P in the same way. Repeating this process recursively, we obtain the complete
k k−1 1
decompositionofP. SinceP ,P ,...,P areshorterthanP,thepathsparticipatinginthecomplete
k k−1 1
decomposition are pyramids and empty paths. Thus we call it the complete decomposition of P into
pyramids.
The complete decomposition of a Dyck path can be also represented by a rooted tree: Given
P =Λk [P ,P ,...,P ],werepresentitbyΛk astherootwithchildrenP ,P ,...,P . Thenwe
k k−1 1 k k−1 1
∗
do the same for P ,P ,...,P and continue recursively, until all the leaves are pyramids or empty
k k−1 1
paths. A Dyck paths is easily restored from its complete decomposition.
4 ANDREIASINOWSKI† ANDTOUFIKMANSOUR‡
An example of complete decomposition is
U4DU2DUDDDU2DDDDUDU2DUD =
=Λ4 [U2DUDD,U2DD, ,UDU2DDUD]
∗ ∅
=Λ4 [Λ2 [UD, ],Λ2, ,Λ1 [U2DDUD]]
∗ ∗ ∅ ∅ ∗
=Λ4 [Λ2 [Λ1, ],Λ2, ,Λ1 [Λ2 [ ,UD]]]
∗ ∗ ∅ ∅ ∗ ∗ ∅
=Λ4 [Λ2 [Λ1, ],Λ2, ,Λ1 [Λ2 [ ,Λ1]]],
∗ ∗ ∅ ∅ ∗ ∗ ∅
and it is illustrated on Figures 2 and 3. On Figure 2 paths appended to a pyramid are shaded in a
more dark colour than the pyramid.
Figure 2. Complete decomposition of a Dyck path.
Λ4
Λ2 Λ2 Λ1
Λ1 Λ2
Λ1
Figure 3. Complete decomposition of a Dyck path represented by a rooted tree.
Note that each Uk in a Dyck path results in a Λk in the complete decomposition. Therefore the
decomposition (2) is valid also for coloured Dyck paths: Pˆ = UˆkDPˆ DPˆ ...DPˆ , and in the
k k−1 1
complete decompositionofacolouredDyckpath, eachUk colouredbyC resultsinΛk colouredbyC.
It is also clear how to restore the coloured Dyck path from its complete decomposition to coloured
pyramids.
We remark that the expression(1) enumerates thus also the following structure: rooted trees with
nedges,eachvertexv colouredbyoneofa colours,whered(v) istheout-degreeofthe vertex. For
d(v)
instance, if each vertex v is coloured by one of C colours, there are 1 3n such trees.
d(v) 2n+1 n
(cid:0) (cid:1)
2.3. A bijection between non-coloured Dyck paths and NCO-trees. We begin with a simple
bijectionϑ: . GivenP , weconstructϑ(P)accordingtothe followingalgorithm. Start
D→NCO ∈D
with the NC-tree which has one point 1 and no edges. Scan P and do the following: For eachU, add
DYCK PATHS WITH COLOURED ASCENTS 5
a new edge beginning in the present point (for a while its end is not determined). For each D, add
the next point, move to it and let it be the end of the last incomplete edge. See Figure 4.
1 1 1 1
5 2 5 2 5 2 5 2
4 3 4 3 4 3 4 3
Figure 4. The function ϑ: .
D →NCO
It is easy to see that ϑ is well defined (since P is a Dyck path, we never need to complete a non-
existing edge, and all edges are completed by the end), and that ϑ is invertible: Given T ,
∈ NCO
scan it from the vertex 1 clockwise. Visiting a vertex, first count in-edges incident with it, and then
out-edges. Foreachin-edgeaddU,foreachout-edgeaddD,andmovetothenextvertex. Itiseasyto
seethatthusobtainedDyckpathP satisfiesϑ(P)=T. Besides,ifP (k)thenϑ(P) (n+1).
∈D ∈NCO
Thus wehaveafamily ofbijections ϑ : (n) (n+1),foralln 0,whichshowsinparticular
n
D →NCO ≥
that (n+1) =C .
n
|NCO |
2.4. Definition of ϕ : D . First we define ϕ for coloured pyramids. Consider Λk C where
D → NC h i
C . We define ϕ(Λk C )=ϑ(C).
∈D h i
Now we define ϕ for all colouredDyck paths. Let Pˆ =Λˆk [Pˆ ,Pˆ ,...,Pˆ ] D. Suppose that
k k−1 1
∗ ∈D
weknowϕ(Λˆk)andϕ(Pˆ)fori=1,2,...,k. Foreachi=1,2,...,k,insertacopyofϕ(Pˆ)intoϕ(Λˆk)
i i
so thatthe vertex1 ofϕ(Pˆ) is mappedto the vertexi+1ofϕ(Λˆk), andthe vertices2,3,... ofϕ(Pˆ)
i i
are mapped clockwise to new vertices between i and i+1 in ϕ(Λˆk) (if Pˆ = nothing happens). See
i
∅
Figure 5.
The function ϕ is invertible. Given T , we want to find Pˆ D such that ϕ(Pˆ) = T. Take
∈ NC ∈ D
the subtree of T with root 1 obtained by recursive adding only out-edges incident with each reached
vertex. After appropriate relabeling of vertices, it forms an NCO-tree V. It corresponds to the base
pyramid Λˆk of Pˆ, where k is equal to the number of edges in V, and the colouring is ϑ−1(V). For
i = 1,2,3,..., the subtree of T attached to the vertex i of V corresponds to Pˆ which is determined
i
recursively. This allows to restore Pˆ.
It is easy to see that if Pˆ D(n) then ϕ(Pˆ) (n+1). Thus we have a family of bijections
ϕ : D(n) (n+1), for∈aDll n 0. ∈ NC
n
D →NC ≥
This completes the proof of Theorem 1.
3. Dyck paths coloured by Dyck paths with ascents of bounded length
In this section we restrictthe Dyck paths used as colours,considering in this role only Dyck paths
with ascents of bounded length.
Let m(n) be the set of Dyck paths of length 2n with ascents of length m. It is known that
M ≤
m(n) is equal to the n-th m-generalized Motzkin number. For m = 2 we have Motzkin numbers
|M |
6 ANDREIASINOWSKI† ANDTOUFIKMANSOUR‡
1 1
3 2 2
1 1
4 2
3 2
3
1
5 2
4 3 1
10 2
9 3
8 4
7 5
6
Figure 5. The function ϕ : D . The bold edges are those corresponding to
D →NC
the base pyramid.
which enumerate Motzkin paths. For m n we have ≥n(n) = C . In this sense the sequence of
n
≥ |M |
sequences of m-generalized Motzkin numbers “converges”, with m , to the sequence of Catalan
→∞
numbers.
Among other structures enumerated by m(n) we have
|M |
The set of rooted trees on n+1 vertices with degree m.
• ≤
The set of all partitions of the vertices of a convex labeled n-polygon to ( m)-sets with
• ≤
disjoint convex hulls.
Denote Mm = m(0), m(1), m(2),... . We consider Mm(n) for fixed m, and prove the
{M M M } D
following:
Theorem 2. There is a bijection between Mm(n) and the set of partitions of the vertices of a labeled
D
convex (2n)-polygon to ( 2m)-sets of even size with disjoint convex hulls. The cardinality of both
sets is n/m−1 (−1)p n−≤mp 3n−mp−p .
p=0 n−mp p n−mp−1
P (cid:0) (cid:1)(cid:0) (cid:1)
DYCK PATHS WITH COLOURED ASCENTS 7
3.1. Enumeration. The generating function A(x) for m(n) satisfies
n≥0
{|M |}
(3) A(x)=1+xA(x)+x2A2(x)+ +xmAm(x).
···
Substituting (1) in (3), we obtain that the generating function h (x) for Mm(n) satisfies
m n≥0
{|D |}
h (x)=1+xh2 (x)+x2h4 (x)+ +xmh2m(x),
m m m ··· m
which is equivalent to
h (x) 1=xh3 (x) (xh2 (x))m+1.
m − m − m
Applying the Lagrange inversion formula on
h (x,a) 1=a xh3 (x,a) (xh2 (x,a))m+1 ,
m − m − m
(cid:0) (cid:1)
we obtain
aℓxℓ ℓ−1 ℓ 3ℓ ℓ (2m 1)j
h (x,a) 1= ( 1)jxmj − ,
m
− ℓ − (cid:18)i(cid:19)(cid:18)j(cid:19)(cid:18) ℓ 1 i (cid:19)
Xℓ≥1 Xi=0Xj=0 − −
which implies that the coefficient of xn in h (x)=h (x,1) is
m m
n/m−1 ( 1)p n mp n−mp−1 3n 3mp 2mp p
[xn](h (x))= − − − − =
m
n mp(cid:18) p (cid:19) (cid:18) i (cid:19)(cid:18)n mp 1 i(cid:19)
Xp=0 − Xi=0 − − −
n/m−1 ( 1)p n mp 3n mp p
(4) = − − − − .
n mp(cid:18) p (cid:19)(cid:18)n mp 1(cid:19)
Xp=0 − − −
This is the cardinality of Mm(n).
D
3.2. Partitions of convex polygons. Denote by m(n) the set of all partitions of the vertices of
E
a convex labeled (2n)-polygon to ( 2m)-sets of even size with disjoint convex hulls. We label the
≤
vertices of the (2n)-polygonby [n] a,b = a ,b ,a ,b ,...,a ,b and depict them appearing on
1 1 2 2 n n
×{ } { }
a circle clockwise in this order, a being the top point. Denote also = m(n).
1 E n≥0,m≥1E
For all n 0, m 1 we shall construct a bijection ρ : Mm(n) Sm(n). It will be presented
n,m
≥ ≥ D → E
as a restriction of a bijection ρ: D .
D →E
We startwith a bijection ψ from to ¯,the setof allpartitionsof the vertices ofa convexlabeled
D E
polygon to sets with disjoint convex hulls. We label the vertices of the polygon by 1,2,3,... and
depict them appearing on a circle clockwise in this order, 1 being the top point.
GivenP weconstructψ(P)accordingtothefollowingalgorithm. Startwiththecirclewithout
∈D
points. Scan P and do the following: For each UkD add the next point which will be the first vertex
of a k-polygon (the other vertices of this polygon are determined later). For each D not preceded by
U, add the next point, move to it and let it be a new vertex of the last incomplete polygon.
Itiseasytoseethatψiswelldefinedandinvertible–similarlytothefunctionϑ. Indeed,comparing
Figures 4 and 6, the reader will easily construct a bijection between and ¯.
NCO E
Now we define ρ: D . First we define it for coloured pyramids.
D →E
Let M . We define ρ(Λk M ) to be a “duplicated ψ(M)”, i. e. for each polygon with vertices
∈ D h i
x ,x ,x ,... in ψ(M), we have a polygon with vertices a ,b ,a ,b ,a ,b ... in ρ(Λk M ).
1 2 3 x1 x1 x2 x2 x3 x3 h i
Now let Pˆ = Λˆk [Pˆ ,Pˆ ,...,Pˆ ] Mm(n), and suppose we know ρ(Λˆk) and ρ(Pˆ) for i =
k k−1 1 i
∗ ∈ D
1,2,...,k. Foreachi=1,2,...,k, inserta copyofρ(Pˆ)into ρ(Λˆk)sothat allthe points ofρ(Pˆ) are
i i
mapped clockwise to new points between a and b . The obtained partition is ρ(Pˆ). See Figure 7 for
i i
an illustration.
8 ANDREIASINOWSKI† ANDTOUFIKMANSOUR‡
1 1 1 1
4 2 4 2 4 2 4 2
3 3 3 3
Figure 6. The function ψ : ¯.
D →E
a1 a1 a1
b3 b1
b2 b1
b1
a3 a2
a1
b2 a2
b4 b1
a4 a2 a1
b8 b1
b3 b2 a8 a2
a3
b7 b2
a7 a3
b6 b3
a6 a4
b5 b4
a5
Figure 7. The function ρ: D .
D →E
The function ρ is invertible. Given T , we want to find Pˆ such that ρ(Pˆ) = T. Consider T
∈ E
as the union of polygons and choose points of T, beginning from a and moving clockwise as follows:
1
from a pass to the vertex connected to it (by an edge of a polygon in the partition), from b pass to
i i
a . Denote by V the union of polygons formed by the chosen points after appropriate relabeling
i+1
of vertices (these polygons have bold edges in Figure 7). It corresponds to the base pyramid of Pˆ
with colouring determined by joining points a and b into one point i and then applying ψ−1. For
i i
DYCK PATHS WITH COLOURED ASCENTS 9
i=1,2,3,...,thepartofT betweenthepointsa andb ofV correspondstoP whicharedetermined
i i i
recursively. This allows to restore Pˆ.
It is easy to see that if Pˆ Mm(n) then ρ(Pˆ) m(n). In particular, each k-ascent in Pˆ results
in a (2k)-polygon in ρ(Pˆ). ∈ThDus we have a famil∈yEof bijections ρ : Mm(n) m(n), for all
n,m
D → E
n 0,m 1, and this completes the proof of Theorem 2.
≥ ≥
3.3. Two special cases. We consider two special cases: m=1 and m=n.
1. Let m=1. Substituting this in (4), we get
n−1( 1)p n p 3n 2p
− − − ,
n p(cid:18) p (cid:19)(cid:18)n p 1(cid:19)
Xp=0 − − −
which is equal to C : since 1(k) =1 for each k, we have M1(n) = (n) =C .
n n
|M | |D | |D |
The partitions of [2n] to even sets with disjoint convex hulls corresponding to the members of
M1(n) in the bijection ρ are those in which each set in partition has two members (all the ways to
D
connect pairs of points of [2n] in convex position by disjoint segments).
2. Let m=n. Substitute this in (4). The only relevant value of p is 0, and we get therefore
1 3n 1 3n
= ,
n(cid:18)n 1(cid:19) 2n+1(cid:18)n(cid:19)
−
which is expected: we have n(n) =C and thus Mn(n)= D(n).
n
|M | D D
The corresponding partitions of [2n] are all possible partitions into even polygons.
4. Dyck paths coloured by Fibonacci paths
In this Section we consider a further restriction of Dyck paths taken as colours.
Let m(n) be the set of Dyck paths of length 2n which have the form Λk1Λk2Λk3... with k m
i
F ≤
– a concatenation of pyramids of length no more than 2m. Note that m(n) m(n). It is known
F ⊂M
that 2(n) is equalto the (n+1)-stFibonacci number; as a generalization m(n) is the (n+1)-st
|F | |F |
m-generalizedFibonacci number (see [10, A092921]). The members of m(n) will be therefore called
F
m-generalized Fibonacci paths. Besides,denote (n)= n(n). Wehave (n) =2n−1 forn>0,and
F F |F |
(0) =1.
|F |
Denote Fm = m, m, m,... and F= , , ,... . We consider Fm(n) for fixed m, and
F(n), and prove{Fth0e fFol1lowFin2g: } {F0 F1 F2 } D
D
Theorem 3. There is a bijection between Fm(n)and the set of diagonal dissections of a labeled
D
convex (n+2)-polygon into 3-, 4-, ...,(m+2)-polygons. The cardinality of both sets is
n−1 1 n+ℓ+1 ℓ+1( 1)i n−1−mi ℓ+1 .
ℓ=0 ℓ+1 ℓ i=0 − ℓ i
P (cid:0) (cid:1)P (cid:0) (cid:1)(cid:0) (cid:1)
4.1. Enumeration. The generating function of the sequence m(n) is
n≥0
{|F |}
1
(n)m xn = .
|F | 1 x x2 xm
nX≥0 − − −···−
Substitutingthisin(1),weobtainthatthegeneratingfunctiong (x)forthesequence Fm(n)
m n≥0
{|D |}
satisfies
1
g (x)=
m 1 xg (x) (xg (x))m
m m
− −···−
which is equivalent to
1 xm(g (x))m
g (x) 1=x(g (x))2 − m .
m m
− 1 xg (x)
m
−
10 ANDREIASINOWSKI† ANDTOUFIKMANSOUR‡
Applying the Lagrange inversion formula on
1 xm(g (x,a))m
g (x,a) 1=ax(g (x,a))2 − m ,
m m
− 1 xg (x,a)
m
−
we obtain
aℓ+1 ℓ+1 ℓ+j ℓ+1 2ℓ+2+j+mi
g (x,a) 1= ( 1)ixℓ+j+mi+1 ,
m
− ℓ+1 − (cid:18) j (cid:19)(cid:18) i (cid:19)(cid:18) ℓ (cid:19)
Xℓ≥0 Xj≥0Xi=0
which implies that the coefficient of xn in g (x)=g (x,1) is
m m
n−1 ℓ+1
1 n+ℓ+1 n 1 mi ℓ+1
(5) ( 1)i − − .
ℓ+1(cid:18) ℓ (cid:19) − (cid:18) ℓ (cid:19)(cid:18) i (cid:19)
Xℓ=0 Xi=0
This is the cardinality of Fm(n).
D
4.2. Dissections of a convex polygon. Denote by (n) the set of all dissections of a labeled
m
R
convex n-polygon into i-polygons with i = 3,4,...,m by non-crossing diagonals. We shall label
the vertices of the n-polygon by α,0,1,...,n clockwise, the top vertex being α. Denote also =
R
(n+2).
n≥0,m≥1Rm+2
S For all n 0,m 1 we construct a bijection σ : Fm(n) (n+2). It will be presented
n,m m+2
≥ ≥ D →R
as a restriction of a bijection σ : F .
D →R
Consider Λk F , a pyramid of length 2k colouredby an generalizedFibonacci path F (k). We
h i ∈F
defineσ(Λk F )tobethedissectionoftheconvexpolygonwithk+2vertices,takingadiagonal(α,i)
h i
if and only if the path F touches the x-axis at point (i,0) (see Figure 8).
α α α α
4 0 4 0 4 0 4 0
3 1 3 1 3 1 3 1
2 2 2 2
Figure 8. Definition of the function σ on pyramids.
Let Pˆ = Λˆk [Pˆ ,Pˆ ,...,Pˆ ] F. Suppose that we know dissections σ(Λˆk) and σ(Pˆ) for
k k−1 1 i
∗ ∈ D
i=1,2,...,k. For eachi=1,2,...,k, inserta copyof σ(Pˆ) into σ(Λˆk)so that the vertexα of σ(Pˆ)
i i
is mapped to the vertex i 1 of σ(Λˆk), the last vertex of σ(Pˆ) is mapped to the vertex i of σ(Λˆk),
i
−
and the vertices 1,2,... of σ(Pˆ) are mapped clockwise to new vertices between i 1 and i of σ(Λˆk).
i
−
After relabeling the vertices we obtain a dissection σ(Pˆ). See Figure 9 for an example.
This function σ invertible: Let T and we want to find Pˆ F such that σ(Pˆ) = T. Let
∈ R ∈ D
V be the union of all the polygons in the dissection that have α as a vertex corresponds, after the
appropriate relabeling of its vertices. V corresponds to the base pyramid of Pˆ which is restored