Fibonacci numbers and positive braids
aa r X i v : . [ m a t h . C O ] M a y FIBONACCI NUMBERS AND POSITIVE BRAIDS
REHANA ASHRAF , BARBU BERCEANU , , AYESHA RIASAT Abstract.
The paper contains enumerative combinatorics for positive braids,square free braids, and simple braids, emphasizing connections with classical Fi-bonacci sequence. The simple subgraph of the Cayley graph of the braid group isanalyzed in the final part. Introduction
The classical Fibonacci sequence, ( F n ) n ≥ : 0 , , , , , . . . appears from time totime in enumerative questions related to Artin braids [2], the geometrical analogueof permutations. The positive n -braids can be defined as words in the alphabet { x , x , . . . , x n − } : ❅❅ x i i − i i + 1 i + 2 n · · · · · · in which we identify two words obtained using finitely many changes of type α ( x i x j ) β ←→ α ( x j x i ) β (for | i − j | ≥ α ( x i x i +1 x i ) β ←→ α ( x i +1 x i x i +1 ) β (for i = 1 , , . . . , n − ❅❅ ❅❅ ❅❅❅❅ ❅❅ ❅❅❅❅ ❅❅ ❅❅❅❅ ≡ ≡ and α α α αβ β β β A central role is played by
Garside braid [10]: ∆ n = x ( x x ) . . . ( x n − . . . x ). keywords and phrases: Positive braids, square free braids, simple braids.This research is partially supported by Higher Education Commission, Pakistan.2010 AMS classification: Primary 11B39, 20F36, 05A15; Secondary 05A05. ❅❅ ❅❅❅❅ ∆ :We will denote by MB n the set of positive braids, by MB + n the set of positive braidsnot containing ∆ n as a subword, and by Div(∆ n ) the set of positive braids whichare subwords of Garside braid:Div(∆ n ) = { ω ∈ MB n | there exist α, β ∈ MB n such that ∆ n = αωβ } . A well known result says that the set of square free positive braids coincides withDiv(∆ n ). In [4] is defined in many ways the set of simple braids , SB n ⊂ Div∆ n .One definition is: Definition 1.1. A simple braid is a positive braid β ∈ MB n which contains a letter x i at most once.In our first computations Fibonacci numbers ( F k ) appear; b k and b + k representsthe number of braids of length k in MB and MB +3 respectively. Theorem 1.2.
The generating function of MB is G MB ( t ) = X k ≥ b k t k = 1 + 2 t + 4 t + 7 t + 12 t + 20 t + . . . where b k = F k +3 − , k ≥ . Theorem 1.3.
The generating function of MB +3 is G MB +3 ( t ) = X k ≥ b + k t k = 1 + 2 t + 4 t + 6 t + 10 t + 16 t + . . . where b + k = 2 F k − , k ≥ . Theorem 1.4.
The number of simple braids in SB n is F n − . The paper contains some other combinatorial problems related to positive braids.In the next section the proofs of the first two theorems are given.In the third section the generating polynomial of the square free braids is com-puted (Proposition 3.1) and the recurrence relation for its coefficients are presented(Proposition 3.2).A proof of Theorem 1.4, the generating polynomial for simple braids, and someproperties of its coefficients (Proposition 4.1) are contained in section 4.The fifth section contains enumerative problems related to the set of conjugacyclasses of simple braids (Proposition 5.1). ibonacci numbers and Positive braids 3
In the last section we analyze the subgraph of the Cayley graph of the braid groupgenerated by simple braids (Proposition 6.2 and Proposition 6.3).Connections between multiple Fibonacci-type recurrence [12] and Jones polyno-mial and Conway-Alexander polynomial for closed braids are presented in [8] and[3]. 2.
Positive braids
The generating function for positive braids was computed by P. Deligne [9] usinginvariants of Coxeter groups. A direct computation for 3-braids was done by P. Xu[14] and an inductive algorithm for G MB n ( t ) and some generalizations are containedin Z. Iqbal [11]. Using any of these references, we have Corollary 2.1. ([9], [14], [11]) The generating function for positive 3-braids is givenby G MB ( t ) = 1(1 − t )(1 − t − t ) · Proof of Theorem 1.2
The expansion in simple parts G MB ( t ) = t − t − t − − t andthe equality (1 − t − t ) − = P k ≥ F k +1 t k gives the result: b k = (2 F k +1 + F k ) − F k +1 + F k +2 ) − F k +3 − . (cid:3) Proof of Theorem 1.3
Every positive braid β can be written in a unique way asa product β = ∆ kn β + with β + ∈ MB + n (see [10]), therefore the decomposition MB = ` k ≥ ∆ k · MB +3 implies: G MB +3 ( t ) = (1 + t + t + . . . ) − · G MB ( t ) = 1 + t + t − t − t = X k ≥ b + k t k . Simple computations shows that b +0 = 1, b +1 = 2 = 2 F , b +2 = 4 = 2 F , and, for k ≥ b + k − b + k − − b + k − = 0, hence the result. (cid:3) For a universal upper bound of the growing type of MB n , see [7].3. Square free braids
To represent an element of
Div (∆ n ), i.e. a positive square free braid, we choosethe canonical form given by the smallest elements in the length-lexicographic order(see [6], [1]): β K,J = β k ,j β k ,j . . . β k s ,j s where β k,j = x k x k − . . . x j +1 x j , 0 ≤ s ≤ n − , ≤ k < k . . . < k s ≤ n − , and j h ≤ k h for h = 1 , . . . , s (the case s = 0 corresponds to the unit β = 1). Forsimplicity, we will write Div n for Div (∆ n ). Let us denote by d n,i the number ofdivisors of ∆ n of length i and by G Div n ( t ) the generating polynomial of the squarefree n -braids. REHANA ASHRAF, BARBU BERCEANU, AYESHA RIASAT
Proposition 3.1. G Div n ( t ) = n ( n − / P i =0 d n,i t i = (1 + t )(1 + t + t ) . . . (1 + t + t + . . . + t n − ) . Proof.
We start the induction with n = 2: Div = { , x } and G Div ( t ) = 1 + t . Thecanonical form of square free braids shows that the map f : Div n − × { , β n − , , β n − , , . . . , β n − ,n − } −→ Div n , defined by f ( ω,
1) = w , f ( ω, β n,k ) = w · β n,k , is a bijection. The generating polyno-mial of the set { , β n − ,k } k =1 ,...,n − is 1 + t + . . . + t n − , so G Div n ( t ) = G Div n − ( t ) · (1 + t + . . . + t n − ). (cid:3) Corollary 3.2.
The sequence ( d n,i ) i =0 ,..., n ( n − is symmetric and unimodular andsatisfies the following recurrence relation:a) d , = 1 , d ,i = 0 if i = 1 ;b) d n +1 ,i = d n,i + d n,i − + . . . + d n,i − n . Simple braids
The canonical form of a simple braids in SB n is β K,J = β k ,j β k ,j . . . β k s ,j s where 1 ≤ k < k < . . . k s ≤ n − j i ≤ k i for all i = 1 , , . . . , s , and also j i +1 > k i for all i = 1 , , . . . , s − SB in the subset of simple braidsof length i in SB n and s n,i its cardinality. The generating polynomial of simple n -braids is denoted by G SB n ( t ) = n − P i =0 s n,i t i . We are interested to count the numberof simple braids G SB n (1). Proposition 4.1.
The sequence ( s n,i ) is given by the recurrence:a) s , = 0 and s ,i = 0 for i = 1 ;b) s n,i = s n − ,i + s n − ,i − + s n − ,i − + . . . + s n − i, . Proof.
The set SB in can be decomposed as disjoint union as follows: SB in = SB in − ∐ ( SB i − n − × { x n − } ) ∐ ( SB i − n − × { x n − x n − } ) ∐ . . . ∐ { x n − . . . x } . (cid:3) Corollary 4.2.
The sequence ( s n,i ) satisfies also the recurrence : s n,i = 2 s n − ,i − + s n − ,i − s n − ,i − . Example 4.3.
Starting with s n, = 1, s n, = n −
1, and using the recurrence ofProposition 4.1 we get s n, = ( n − n + 2) / s n, = ( n − n + 4)( n − /
3! and s n, = ( n − n + 1)( n + 5 n − / ibonacci numbers and Positive braids 5 last non zero coefficient is s n,n − = 2 n − (if n ≥ s n,i (on the n -thline) are given in the triangle: 11 11 2 21 3 5 41 4 9 12 8 Proposition 4.4. s n,i is a polynomial in n of degree i and leading coefficient is /i ! .Proof. The induction by i starts with s n, = 1 and s n, = n −
1. Using Proposition4.1, we have s n,i − s n − ,i = s n − ,i − + s n − ,i − + . . . + s n − i, where the sum is a polynomial in n of degree i − / ( i − s n,i is a polynomial in n of degree i and leading coefficient is 1 /i !. (cid:3) Proof of Theorem 1.4
By definition G SB n (1) = s n, + s n, + s n, + s n,n − + . . . + s n,n − .Using the recurrence given in Proposition 4.1, we expand G SB n (1) = n − P i =0 s n,i and get G SB n (1) = 2 G SB n − (1) + G SB n − (1) + G SB n − (1) + . . . + G SB (1) + G SB (1) . Starting an induction with G SB (1) = 1 = F = F , G SB (1) = 2 = F , G SB (1) =5 = F , we obtain G SB n (1) = 2 F n − + F n − + . . . + F + F + F = 2 F n − + F n − + . . . + F + F = · · · = 2 F n − + F n − + F n − = 2 F n − + F n − = F n − + F n − = F n − . (cid:3) Conjugacy classes of simple braids
A simple braid β ∈ SB n is conjugate to the braid β A = ( x x . . . x s − )( x s +1 . . . x s − ) . . . ( x s r − +1 . . . x s r − )(i.e. there is a positive braid α ∈ MB n such that βα = αβ A ); here A = ( a , a , . . . , a r )is a sequence of integers satisfying a ≥ a ≥ . . . ≥ a r ≥ s i = a + a + . . . + a i .Conversely, if β A and β A ′ are conjugate, then the sequences A and A ′ coincide (see[4]). REHANA ASHRAF, BARBU BERCEANU, AYESHA RIASAT
The generating polynomial for distinct conjugate n -simple braid is denoted by Cf n ( t ) = n − P i =0 c n,i t i , where c n,i is the number of conjugacy classes of positive simplebraids of length i . A partition of a positive integer m is a representation of m in a form m = m + m + . . . + m k where the integers m , m , . . . , m k satisfy theinequalities m ≥ m ≥ . . . ≥ m k ≥
1. The number of partitions of m into k partsis denoted by P ( m, k ) (see [13]). Proposition 5.1.
The number of conjugacy classes of simple braids of length i isgiven by c n,i = P ( i + min ( i, n − i ) , min ( i, n − i )) . Proof.
Consider β A = ( x x . . . x s − )( x s +1 . . . x s − ) . . . ( x s r − +1 . . . x s r − ), the canon-ical representative of a conjugacy class in SB n , of length i = s r − r .We asso-ciate to the sequence A = ( a ≥ a ≥ . . . a r ) (here a r ≥
2) the partition of i , i = ( a −
1) + ( a −
1) + . . . ( a r − s r = a + a + . . . + a r ≤ n implies i + r ≤ r , therefore the number of conjugacy classes of simple braids of length i isgiven by the number of partitions of i into at most n − i parts: c n,i = P ( i,
1) + P ( i,
2) + . . . + P ( i, min ( i, n − i )) . Using the relation P ( n + k, k ) = k P i =1 P ( n, k ) ( k ≤ n ) (see [13]), we obtain theresult. (cid:3) Simple Graph
We consider the subgraph of the Cayley graph of the group B n with vertices thesimple braids. Definition 6.1.
The simple graph Γ SB n is the graph with vertices SB n and edgesbetween the simple braids β ↔ βx i ( i = 1 , . . . , n − SB n is F n − . Proposition 6.2.
The number of edges of the graph Γ SB n is: e (Γ SB n ) = ( n − s n, + ( n − s n, + . . . + s n,n − . Proof.
The number of edges between a vertex β of length i and vertices of length i + 1, βx j , is the number of letters x j which are not in the simple braid β , and thisnumber is n − − i ; this gives ( n − − i ) s n,i edges between simple braids of length i and i + 1. (cid:3) Proposition 6.3. a) The graph Γ SB n is connected and n -partite.b) Γ SB n is planar if and only if n ≤ . ibonacci numbers and Positive braids 7 Proof. a) Any vertex β = x i x i . . . x i k is connected to the empty word 1 by thepath 1— x i — x i x i — . . . — β . The parts are given by simple braids of the samelength. More generally, the Cayley graph of Coxeter groups are multipartite becausemultiplication by generators modifies the length of words with ± SB n is canonically embedded in Γ SB n +1 . The graph Γ SB contains K , as a subgraph and Γ SB has a planar imbedding, see the next pictures: • e • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • ✟✟✟✟✟✟❍❍❍❍❍❍ ✟✟✟✟✟✟❍❍❍❍❍❍ ✟✟✟✟✟✟✟✟✟✟✟✟❍❍❍❍❍❍❍❍❍❍❍❍ ✟✟✟✟✟✟✟✟✟❍❍❍❍❍❍❍❍❍ ✟✟✟✟✟✟✟✟✟✟✟✟❍❍❍❍❍❍❍❍❍❍❍❍ ✟✟✟✟✟✟✟✟✟✟✟✟❍❍❍❍❍❍❍❍❍❍❍❍ ✟✟✟✟✟✟✟✟❍❍❍❍❍❍❍❍ ❍❍❍❍❍❍❍❍❍❍✟✟✟✟✟✟✟✟✟✟ ❍❍❍❍❍❍❍❍❍❍❍❍✟✟✟✟✟✟✟✟✟✟✟✟ ❍❍❍❍❍❍❍❍❍❍❍✟✟✟✟✟✟✟✟✟✟✟ ❍❍❍❍❍❍❍❍❍❍❍❍❍❍✟✟✟✟✟✟✟✟✟✟✟✟✟✟ ❍❍❍❍❍❍❍❍❍❍❍❍❍❍✟✟✟✟✟✟✟✟✟✟✟✟✟✟ ❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟✟ ✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁✁ ❆❆❆❆❆❆❆❆❆❆❆❆❆❆❆❆❆❆❆❆ ✟✟❍❍ ✁✁✁✁✁✁✁✁✁✁✁✁✁ ❆❆❆❆❆❆❆❆❆❆❆❆❆✑✑✑✑✑✑✑✑✑✑✑✑ ◗◗◗◗◗◗◗◗◗◗◗◗✑✑✑✑✑✑✑✑✑✑✑✑ ◗◗◗◗◗◗◗◗◗◗◗◗(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) ❅❅❅❅❅❅❅❅✘✘✘ ❳❳❳(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) ❅❅❅❅❅❅✡✡✡✡ ❏❏❏❏(cid:0)(cid:0) ❅❅(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) ❅❅❅❅❅❅❆❆❆❆❆❆✁✁✁✁✁✁ ❆❆❆❆❆❆✁✁ ❆❆❆❆ ✁✁❆❆❆❆❆❆ ✁✁✁✁✁✁❆❆ ✁✁❅❅❅❅ (cid:0)(cid:0)(cid:0)(cid:0)❆❆❆❆ ✁✁✁✁❩❩❩❩ ✚✚✚✚✁✁ ❆❆❅ (cid:0)❍❍ ✟✟✁✁ ✁✁ ❆❆ ❆❆✏✏✏✏✏✏ ✏✏✏✏✏✏ PPPPPPPPPPPP(cid:0)(cid:0) ❅❅❳❳❳❳ ✘✘✘✘❆❆ ✁✁✭✭✭✭✭✭ ❤❤❤❤❤❤✁✁✁✁✁✁ Γ SB REHANA ASHRAF, BARBU BERCEANU, AYESHA RIASAT ★★★★★★★★★★✦✦✦✦✦✦✦✦✦✦✦✦✦✦✦✦✦✦✦✦★★★★★★★★★★❝❝❝❝❝❝❝❝❝❝ ❝❝❝❝❝❝❝❝❝❝ ❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛❛ • e • • • • • • • • • • • • • • • • A K , subgraph of Γ SB (cid:3) The close relations between simple braids in SB n and the corresponding permu-tations in the symmetric group Σ n and also the simple part of the permutahedron(the simple graph Γ SB n is its one dimensional skeleton) are studied in [5]. References [1] U. Ali, B. Berceanu,
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