aa r X i v : . [ m a t h . C O ] D ec Descent pattern avoidance
Richard Ehrenborg and JiYoon Jung
Abstract
We extend the notion of consecutive pattern avoidance to considering sums over all permu-tations where each term is a product of weights depending on each consecutive pattern of afixed length. We study the problem of finding the asymptotics of these sums. Our technique isto extend the spectral method of Ehrenborg, Kitaev and Perry. When the weight depends onthe descent pattern we show how to find the equation determining the spectrum. We give twolength 4 applications. First, we find the asymptotics of the number of permutations with notriple ascents and no triple descents. Second, we give the asymptotics of the number of permu-tations with no isolated ascents or descents. Our next result is a weighted pattern of length 3where the associated operator only has one non-zero eigenvalue. Using generating functions weshow that the error term in the asymptotic expression is the smallest possible.
Ehrenborg, Kitaev and Perry [3] used the spectrum of linear operators on the space L ([0 , m )to study the asymptotics of consecutive pattern avoidance. We extend their techniques to studyasymptotics of sums over all permutations where each term is a product of weights which depend onthe consecutive patterns of a fixed length m + 1. When the weights are all zero or one, this reducesto studying consecutive pattern avoidance. Furthermore, when the weights depend on the descentpattern, we show how to obtain the equation whose roots are the spectrum of the associated linearoperator. In general this is a transcendental equation.We give two length 4 examples. First we study the number of permutations with no tripleascents and no triple descents. This is equivalent to { , } -avoiding permutations. Wedetermine the transcendental eigenvalue equation and a numerical approximation to the largestroot, which gives the asymptotics of the number such permutations.The second example is permutations that avoid the ten alternating patterns 1324, 1423, 2314,2413, 3412 and 2143, 3142, 3241, 4132, 4231. This is the class of permutations with no isolatedascents or descents. Yet again, we obtain the transcendental eigenvalue equation satisfied by thespectrum and give numerical approximation to its largest root.We next turn to a weighted length 3 example. We are interested in the sum over all 123-avoiding permutations where the term is 2 to the power of the number of double descents. Herewe also consider the extra conditions if the permutation begins/ends with an ascent or a descent.The associated operator only has one non-zero eigenvalue, namely 1. Hence the asymptotics is aconstant c times n factorial and the error term is bounded by n ! · r n where r is an arbitrary smallpositive number.It remains to understand the error term. We are able to find the associated generating functions.Furthermore, we show that the error term is the smallest possible! The asymptotics is c · n ! (wherethe constant c is irrational, in fact, transcendental) and the explicit expression is the nearest integer1o c · n ! for large enough n . This behavior also occurs with the derangement numbers. This classicalsequence makes its appearance as one of the sequences that we study.We end the paper with concluding remarks and open problems. For x , x , . . . , x k distinct real values, define Π( x , x , . . . , x k ) to be the unique permutation σ inthe symmetric group S k such that x i < x j if and only if σ i < σ j for all indices 1 ≤ i < j ≤ k . Wesay that a permutation π in S n consecutively avoid a permutation σ in S m if there is no index i such that Π( π i , π i +1 , . . . , π i + m − ) = σ .Let wt be a real-valued weight function on the symmetric group S m +1 . Similarly, let wt , wt be two real-valued weight functions on the symmetric group S m . We call wt and wt the initial,respectively, the final weight function. We extend these three weight functions to the symmetricgroup S n for n ≥ m by definingWt( π ) = wt (Π( π , π , . . . , π m )) · n − m Y i =1 wt(Π( π i , π i +1 , . . . , π i + m )) · wt (Π( π n − m +1 , π n − m +2 , . . . , π n )) . In other words, the weight of a permutation π in S n is the product of the initial weight functionwt applied to the m first entries of π with the product of the weight function wt applied to everysegment of π of length m + 1 with the final weight function wt applied to the m last entries of π .The question is what can one say about the quantity α n = X π ∈ S n Wt( π ) . Consecutive pattern avoidance can be studied this way by using the weight functions wt ( σ ) =wt ( σ ) = 1 for all σ in S m and wt( σ ) = 1 if σ S and wt( σ ) = 0 otherwise, where S ⊆ S m +1 isthe set of forbidden patterns. Observe then that a permutation π ∈ S n avoids the patterns in S if and only if Wt( π ) = 1. Note that by letting the initial weight function wt and the final weightfunction wt be 0 , L ([0 , m ) naturally extend to this moregeneral setting of weights on permutations.Define the function χ on the ( m + 1)-dimensional unit cube [0 , m +1 by χ ( x ) = wt(Π( x )). Notethat χ is undefined on a point with two equal coordinates. However, this situation occurs on a setof measure zero and hence can be ignored. Next define the operator T on the space L ([0 , m ) by T ( f ( x , . . . , x m )) = Z χ ( t, x , . . . , x m ) · f ( t, x , . . . , x m − ) dt. (2.1)Note that L ([0 , m ) is a Hilbert space with the inner product defined by( f, g ) = Z [0 , m f ( x , . . . , x m ) · g ( x , . . . , x m ) dx · · · dx m . T ∗ is defined by the relation ( f, T ∗ ( g )) = ( T ( f ) , g ). For the operator T definedin equation (2.1) we have that T ∗ ( f ( x , . . . , x m )) = Z χ ( x , . . . , x m , u ) · f ( x , . . . , x m , u ) du Finally, the spectrum of an operator T is all the values λ such that T − λ · I is not an invertibleoperator.Similarly to the function χ , define the two functions κ and µ on the m -dimensional unit cube[0 , m by κ ( x ) = wt (Π( x )) and µ ( x ) = wt (Π( x )).Generalizing the main result in [3], we have the following theorem. Theorem 2.1.
The non-zero spectrum of the associated operator T consists of discrete eigenvaluesof finite multiplicity which may accumulate only at . Furthermore, let r be a positive real numbersuch that there is no eigenvalue of T with modulus r and let λ , . . . , λ k be the eigenvalues of T greaterin modulus than r . Assume that λ , . . . , λ k are simple eigenvalues with associated eigenfunctions ϕ i and that the adjoint operator T ∗ has eigenfunctions ψ i corresponding the eigenvalues λ i . Then wehave the expansion α n /n ! = (cid:0) T n − m ( κ ) , µ (cid:1) = k X i =1 ( ϕ i , µ ) · (cid:0) κ, ψ i (cid:1)(cid:0) ϕ i , ψ i (cid:1) · λ n − mi + O ( r n ) . (2.2)The proof is the same as in [3, Section 2.2] and hence omitted.Theorem 2.1 requires us to determine both the eigenfunction ϕ and the adjoint eigenfunction ψ for each eigenvalue in order to compute the constant in each term. However, when the weight func-tion has symmetry in the sense described below then the adjoint eigenfunction can be determinedfrom the eigenfunction.Let J be the involution on L ([0 , m ) given by J ( f ( x , x , . . . , x m )) = f (1 − x m , . . . , − x , − x ). Note that J is a self-adjoint operator on L ([0 , m ), that is, ( J f, g ) = ( f, J g ). Similar to [3,Lemma 4.7] we have that
Lemma 2.2.
Assume that the weight function wt is real-valued and satisfies the symmetry wt( σ ) = wt( m + 2 − σ m +1 , m + 2 − σ m , . . . , m + 2 − σ ) for all σ ∈ S m +1 . If ϕ is an eigenfunction of the operator T with eigenvalue λ then ψ = J ϕ is an eigenfunction of the adjoint T ∗ with the eigenvalue λ . Furthermore, we have the equality (cid:0) f, ψ (cid:1) = ( ϕ, J f ) for a real-valued function f . To prove Lemma 2.2, the only part that differs from the proof in [3, Lemma 4.7] is the line (cid:0) f, ψ (cid:1) = (cid:0) f, J ϕ (cid:1) = ( f, J ϕ ) = ( J f, ϕ ) = (cid:0) ϕ, J f (cid:1) = ( ϕ, J f ). We now introduce weighted descent pattern avoidance and the connection with consecutive patternavoidance. For a permutation π = π π · · · π n ∈ S n define its descent word (see for instance [4, 7])3o be u ( π ) = u u · · · u n − where u i = a if π i < π i +1 and u i = b if π i > π i +1 , that is, an a atposition i encodes that π has an ascent at position i and a b encodes a descent.Let wt be a weight function on ab -words of length m , that is, the set { a , b } m . Similarly, letwt and wt be weight functions on ab -words of length m −
1. We extend this weight function towords of length n greater than m − v · · · v n ) = wt ( v · · · v m − ) · n − m +1 Y i =1 wt( v i · · · v i + m − ) · wt ( v n − m +2 · · · v n ) . Finally, we extend the weight to permutations by letting Wt( π ) = Wt( u ( π )).Recall that the word x has the word w as a factor if we can write x = v · w · z , where v and z are also words and the dot denotes concatenation. Let U be a collection of ab -words of length m ,that is, U is a subset of { a , b } m . Define S ( U ) by S ( U ) = { σ ∈ S m +1 : u ( σ ) ∈ U } . It is clear that a permutation π that avoids the descent patterns in U is equivalent to that thepermutation avoids the consecutive patterns in S ( U ). Hence descent pattern avoidance is a specialcase of consecutive pattern avoidance.A few examples are in order. Example 3.1. m = 1 and U = { b } . There is only one permutation without any descents, namely12 · · · n , and hence α n = 1. Example 3.2. m = 2 and U = { ab } . This forces the permutation to have no peaks. Hence α n = 2 n − for n ≥ Example 3.3. m = 2 and U = { aa , bb } . This forces the permutation to be alternating. Alternat-ing permutations are enumerated by the Euler numbers, that is, α n = 2 · E n for n ≥ α n = 1for n ≤
1. See for instance [7, Section 1.6.1] or [3, Example 1.11].For an ab -word u of length m − P u to be the subset of the unitcube [0 , m corresponding to all vectors with descent word u , that is, P u = { ( x , . . . , x m ) ∈ [0 , m : x i ≤ x i +1 if u i = a and x i ≥ x i +1 if u i = b } . Observe that the unit cube [0 , m is the union of the 2 m − descent polytopes. Similar to [3,Proposition 4.3 and Corollary 4.4] we have the next proposition. Furthermore, the proof is alsosimilar and hence omitted. Proposition 3.4.
Let T be the operator associated with a weighted descent pattern avoidance and k an integer such that ≤ k ≤ m − . Let u be an ab -word of length m − and f a function in L ([0 , m ) . Then the function T k ( f ) restricted to the descent polytope P u only depends on thevariables x through x m − k . A direct consequence of Proposition 3.4 is that the eigenfunctions have a special form:
Corollary 3.5. If ϕ is an eigenfunction of T associated to a non-zero eigenvalue, then the eigen-function ϕ restricted to any descent polytope P u only depends on the variable x . V be the subspace of L ([0 , m ) consisting of all functions f such that the restriction f | P u only depends on the variable x for all words u of length m −
1. Let f be a function in thesubspace V . Then the function T ( f ) is described as follows. For an ab -word u of length m − y ∈ { a , b } we have T ( f ) | P uy = Z x wt( a uy ) · f ( t ) | P a u dt + Z x wt( b uy ) · f ( t ) | P b u dt. (3.1)In light of Corollary 3.5 to solve the eigenvalue problem for the operator T : L ([0 , m ) −→ L ([0 , m ), it is enough to solve the eigenvalue problem for the restricted operator T | V : V −→ V .The restricted operator is of a particular form, which we describe in the next section. Recall that for a square matrix M the exponential matrix of M is defined by the converging powerseries e M = X k ≥ M k /k ! = I + M + M / M /
3! + · · · . The general solution of the system of first order linear equations ddx ~p ( x ) = M · ~p ( x ) is given by ~p ( x ) = e M · x · ~c where ~c is the initial condition ~p (0).Let γ ( M ) denote the matrix γ ( M ) = Z e M · t dt, where the integration is entrywise. Observe that M · γ ( M ) = Z M · e M · t dt = (cid:2) e M · t (cid:3) = e M − I. (4.1)Hence when M is non-singular we can write γ ( M ) = M − · ( e M − I ). Also note that by integratingthe power series of e M · t term by term we obtain that γ ( M ) = X k ≥ M k / ( k + 1)! = I + M/ M /
3! + M /
4! + · · · . Lemma 4.1.
The two following indefinite integrals hold: Z e M · t dt = γ ( M · t ) · t + ~C, Z M · t · e M · t dt = t · e M · t − γ ( M · t ) · t + ~C. Proof.
The first identity follows by integrating the power series termwise. The second identityfollows from integrating the equality M · t · e M · t + e M · t = ddt (cid:0) t · e M · t (cid:1) .Let A and B be two k × k matrices. Consider the integral operator T defined on vector-valuedfunctions by T ( ~p ( x )) = A · Z x ~p ( t ) dt + B · Z x ~p ( t ) dt, (4.2)5here the integration is componentwise.Observe that the restricted operator described in equation (3.1) is of the form (4.2) by letting A and B be matrices indexed by ab -words of length m − A uy, a u = wt( a uy ) and B uy, b u = wt( b uy )where y ∈ { a , b } and u is an ab -word of length m −
2, and the remaining entries of the matricesare 0.The following theorem concerns the eigenvalues and eigenfunctions of the operator in (4.2).
Theorem 4.2.
The non-zero spectrum of the operator T is given by the set of non-zero roots ofthe equation det( P ) = 0 , where the matrix P is given by P = − λ · I + B · γ (( A − B ) /λ ) , (4.3) and the eigenfunctions are of the form ~p ( x ) = e ( A − B ) /λ · x · ~c , where the vector ~c satisfies the equation P · ~c = 0 .Proof. Differentiate the eigenfunction equation λ · ~p = T ( ~p ) with respect to x to obtain the differ-ential equation ddx ~p ( x ) = M · ~p ( x ) , where we let M denote the matrix 1 /λ · ( A − B ). This equation has the solution ~p ( x ) = e M · x · ~c, where ~c is the initial condition. Substituting the solution for the differential equation back into theeigenfunction equation, we obtain λ · e M · x · ~c = A · Z x e M · t · ~c dt + B · Z x e M · t · ~c dt = A · [ γ ( M · t ) · t ] x · ~c + B · [ γ ( M · t ) · t ] x · ~c = (( A − B ) · γ ( M · x ) · x + B · γ ( M )) · ~c = (cid:0) λ · (cid:0) e M · x − I (cid:1) + B · γ ( M ) (cid:1) · ~c. Canceling terms we obtain P · ~c = 0. We can only find the non-zero vector ~c if the matrix P issingular, that is, has a zero determinant.In the case when A − B is non-singular the condition in Theorem 4.2 can be expressed as0 = det( P ) · det( M )= det (cid:16) − A + B · e ( A − B ) /λ (cid:17) . Theorem 4.3.
An eigenvalue λ of the operator T is simple if its associated eigenfunction ~p ( x ) satisfies the vector identity B · e ( A − B ) /λ · ~p (0) = 0 . (4.4)6 roof. Assume that the eigenvalue λ is not simple, that is, it satisfies the generalized eigenvalueequation λ · ~q = T ( ~q ) + ~p . Differentiate this equation to obtain λ · ddx ~q ( x ) = ( A − B ) · ~q ( x ) + ddx ~p ( x ) . Again let M = ( A − B ) /λ . Multiply both sides with 1 /λ · e − M · x to obtain e − M · x · ddx ~q ( x ) − M · e − M · x · ~q ( x ) = 1 /λ · e − M · x · ddx ~p ( x ) . This equation is equivalent to ddx (cid:0) e − M · x · ~q ( x ) (cid:1) = 1 /λ · M · ~c. Hence we have the general solution ~q ( x ) = 1 /λ · e M · x · M · ~c · x + e M · x · ~d, where ~d is a constant vector. Without loss of generality we can set ~d = 0 since we are lookingfor a particular solution. Inserting the particular solution 1 /λ · e M · x · M · ~c · x into the generalizedeigenvalue equation, we obtain M · x · e M · x · ~c = A/λ · Z x M · t · e M · t dt · ~c + B/λ · Z x M · t · e M · t dt · ~c + e M · x · ~c = A/λ · (cid:2) t · e M · t − γ ( M · t ) · t (cid:3) x · ~c + B/λ · (cid:2) t · e M · t − γ ( M · t ) · t (cid:3) x · ~c + e M · x · ~c = M · (cid:0) x · e M · x − γ ( M · x ) · x (cid:1) · ~c + B/λ · (cid:0) e M − γ ( M ) (cid:1) · ~c + e M · x · ~c. Canceling terms using the identity (4.1) and multiplying by λ we have0 = B · (cid:0) e M − γ ( M ) (cid:1) · ~c + λ · ~c. Adding the equation P · ~c = 0 to this identity gives us the conclusion of the theorem. Let us consider the case when we avoid the two words aaa and bbb . This is equivalent to avoidingthe consecutive patterns 1234 and 4321. In this case we have the two matrices A = and B = . Note the matrix A − B is invertible and diagonalizable. To simplify calculations let τ = s √
52 and σ = s − √ . A − B are ± σ and ± τ · i .Using a computer algebra package as Maple, we obtain that the determinant of the matrix P from Theorem 4.2 expands as20 λ · det( P ) = 8 + (cid:16) i + √ · ( τ + σ · i ) (cid:17) · e ( σ + τ · i ) /λ + (cid:16) − i + √ · ( τ − σ · i ) (cid:17) · e ( σ − τ · i ) /λ + (cid:16) − i + √ · ( − τ + σ · i ) (cid:17) · e ( − σ + τ · i ) /λ + (cid:16) i + √ · ( − τ − σ · i ) (cid:17) · e ( − σ − τ · i ) /λ . Thus we obtain
Proposition 5.1.
Let λ be the largest real positive root of the equation − (cid:16) i + √ · ( τ + σ · i ) (cid:17) · e ( σ + τ · i ) /λ + (cid:16) − i + √ · ( τ − σ · i ) (cid:17) · e ( σ − τ · i ) /λ + (cid:16) − i + √ · ( − τ + σ · i ) (cid:17) · e ( − σ + τ · i ) /λ + (cid:16) i + √ · ( − τ − σ · i ) (cid:17) · e ( − σ − τ · i ) /λ . (5.1) Then λ is the largest eigenvalue (in modulus) of the associated operator T and the asymptotics ofthe number of permutations without triple ascents and triple descents is given by α n /n ! = c · λ n − + O ( r n ) , where c and r are two positive constants such that r < λ .Proof. It remains to show that the eigenvalue λ is simple. Observe that the de Bruijn graph withthe two directed edges aa aaa −→ aa and bb bbb −→ bb removed is ergodic. Now the conclusion followsfrom combining Theorems 1.7 and 4.2 in [3].Solving equation (5.1) numerically we obtain the three largest roots: λ = 0 . . . .λ , = − . . . . ± . . . . · i. Hence we have that r is bounded below by | λ , | = 0 . . . . .For the eigenvalue λ = 0 . . . . we can solve for the vector ~c and we have ~c = . . . . . . . . . . . . . ϕ = e ( A − B ) /λ · x · ~c and adjoint eigenfunction ψ = J ϕ . Notethat when we restrict the adjoint eigenfunction ψ to a descent polytope we obtain a function onlydepending on the last variable x . For these two functions we calculate( ϕ, ) = (cid:0) , ψ (cid:1) = 0 . . . . , (cid:0) ϕ, ψ (cid:1) = 0 . . . . . Combining this we have the constant( ϕ, ) · (cid:0) , ψ (cid:1)(cid:0) ϕ, ψ (cid:1) = 0 . . . . . Thus in numerical terms we have that the asymptotics for the number of permutations with notriple ascents and triple descent is given by0 . . . . · (0 . . . . ) n − · n ! . We next consider the case when we avoid the two words aba and bab . This is equivalent to avoidingthe ten alternating permutations 1324, 1423, 2314, 2413, 3412 and 2143, 3142, 3241, 4132, 4231.In this case we have the two matrices A = and B = . Yet again the matrix A − B is invertible and diagonalizable. The eigenvalues are ± τ and ± σ · i .Similar to Proposition 5.1 we have: Proposition 5.2.
Let λ be the largest real positive root of the equation − (cid:16) − i + √ · ( − τ + σ · i ) (cid:17) · e ( τ + σ · i ) /λ + (cid:16) i + √ · ( − τ − σ · i ) (cid:17) · e ( τ − σ · i ) /λ + (cid:16) i + √ · ( τ + σ · i ) (cid:17) · e ( − τ + σ · i ) /λ + (cid:16) − i + √ · ( τ − σ · i ) (cid:17) · e ( − τ − σ · i ) /λ . (5.2) Then λ is the largest eigenvalue (in modulus) of the associated operator T and the asymptotics ofthe number of permutations not having any isolated ascents or descents is given by α n /n ! = c · λ n − + O ( r n ) , where c and r are two positive constants such that r < λ . λ is simple. The onlydifference is that we consider the de Bruijn graph with the two edges ab aba −→ ba and ba bab −→ ab removed.Numerically, we find the following three largest roots to equation (5.2): λ = 0 . . . . ,λ , = 0 . . . . ± . . . . · i. The next largest root λ , bounds r from below by | λ , | = 0 . . . . .Similar to Subsection 5.1 we can obtain the numerical asymptotic expression for the quantity α n .The numerical data is as follows: ~c = . . . . . . . . . . . . , and ( ϕ, ) = (cid:0) , ψ (cid:1) = 0 . . . . , (cid:0) ϕ, ψ (cid:1) = 0 . . . . . Combining this we have the constant( ϕ, ) · (cid:0) , ψ (cid:1)(cid:0) ϕ, ψ (cid:1) = 0 . . . . . Finally, we conclude that the asymptotics for the number of permutations with no isolated ascentsand no isolated descents is given by0 . . . . · (0 . . . . ) n − · n ! . Define a weight function on the set of ab -words of length 2 such that wt( aa ) = 0, wt( bb ) = 2 andwt( ab ) = wt( ba ) = 1 and the initial and final weight functions wt and wt are identical to 1. Weare interested in understanding the sum α n = X π ∈ S n Wt( π ) . A more explicit way to write this sum is as follows α n = X π bb ( π ) , where the sum is over all 123-avoiding permutations of length n and bb ( π ) denotes the number ofdouble descents of π . 10et us refine the number α n by considering if the permutation begins with an ascent or adescent, and similarly how the permutation ends, that is, we define α n ( a , a ), α n ( a , b ), α n ( b , a ) and α n ( b , b ) for n ≥ α n ( x, y ) = X Wt( π ) , where the sum is over all permutations π in S n whose descent word u ( π ) begins with the letter x and ends with the letter y . Note that α ( x, y ) is given by the Kronecker delta δ x,y . These quantitiescan also be expressed by changing the initial and final weight functions.By the symmetry π , π , . . . , π n n + 1 − π n , . . . , n + 1 − π , n + 1 − π we have that α n ( a , b ) = α n ( b , a ).First we consider the spectrum of the associated operator. Theorem 6.1.
The only non-zero eigenvalue of the operator T is λ = 1 . This is a simple eigen-value. Furthermore, the eigenfunction ϕ and the adjoint eigenfunction ψ associated with this eigen-value are given by ϕ = e − x · (cid:26) − x if ≤ x ≤ y ≤ , − x if ≤ y ≤ x ≤ , and ψ = e y − · (cid:26) y if ≤ x ≤ y ≤ ,y + 1 if ≤ y ≤ x ≤ . Proof.
The associated operator T can be written in the form (4.2) using the matrices A = (cid:18) (cid:19) and B = (cid:18) (cid:19) . Note that A − B has eigenvalue − A − B consists of one Jordan block of size 2. Computing the matrix P weobtain 0 = det( P ) = exp ( − /λ ) · λ · ( λ − , which only has the non-zero root λ = 1. Furthermore for this root, the null space of the matrix P is spanned by the vector ~c = (cid:18) (cid:19) . Finally, it is straightforward to verify B · e M · ~c = ~
0, hence λ = 1 is a simple eigenvalue byTheorem 4.3. Moreover the eigenfunction ϕ is given by ϕ = exp (cid:18)(cid:18) − − (cid:19) · x (cid:19) · (cid:18) (cid:19) = e − x · (cid:18) − x − x (cid:19) . Since the weight function wt satisfies the symmetry in Lemma 2.2, we obtain that the adjointeigenfunction is given by ψ = J ( ϕ ). Theorem 6.2.
The asymptotics of the sequences α n ( a , a ) , α n ( a , b ) , α n ( b , b ) and α n are given by α n ( a , a ) /n ! = e − /e + O ( r n ) ,α n ( a , b ) /n ! = 1 − /e + O ( r n ) ,α n ( b , b ) /n ! = 1 /e + O ( r n ) ,α n /n ! = e − /e + O ( r n ) , where r is an arbitrary small positive real number. roof. Let a denote the function encoding an ascent, that is, a ( x, y ) = 1 if x < y and 0 otherwise.Similarly, let b be the function encoding a descent, that is, b ( x, y ) = 1 if x > y and 0 otherwise.Note that we have that J a = a and J b = b . By letting the initial function κ and the finalfunction µ vary over the two functions a and b , we obtain the constant term in the asymptoticexpression in Theorem 2.1. First we compute the inner products( ϕ, a ) = (cid:0) a , ψ (cid:1) = 1 − /e, ( ϕ, b ) = (cid:0) b , ψ (cid:1) = 1 /e, (cid:0) ϕ, ψ (cid:1) = 1 /e, where we used Lemma 2.2 for two of the five equalities. Hence the constants are:( ϕ, a ) · (cid:0) a , ψ (cid:1)(cid:0) ϕ, ψ (cid:1) = e − /e, ( ϕ, b ) · (cid:0) a , ψ (cid:1)(cid:0) ϕ, ψ (cid:1) = 1 − /e, ( ϕ, b ) · (cid:0) b , ψ (cid:1)(cid:0) ϕ, ψ (cid:1) = 1 /e. This proves the three first results of the theorem. The fourth result is obtained by adding theasymptotic expressions for α n ( a , a ), α n ( a , b ), α n ( b , a ) and α n ( b , b ).In order to study these sequences further, we introduce the associated exponential generatingfunctions. Let F x,y ( z ) denote the generating function F x,y ( z ) = X n ≥ α n ( x, y ) · z n n ! . Similarly, let F ( z ) be the generating function for the sequence α n . Proposition 6.3.
The generating function F x,y ( z ) satisfies the following equation: F x,y ( z ) = δ x,y · z
2! + δ x, b · δ y, a · · z Z z ( F x, a ( w ) + 2 · F x, b ( w )) · F b ,y ( w ) dw + δ x, a · Z z F b ,y ( w ) dw + δ x, b · Z z w · F b ,y ( w ) dw + δ y, b · Z z ( F x, a ( w ) + 2 · F x, b ( w )) dw + δ y, a · Z z ( F x, a ( w ) + 2 · F x, b ( w )) · w dw. (6.1)12 roof. We demonstrate that all the terms on the right-hand side are in fact counting permutations.The first term corresponds to permutations of length 2. The second term corresponds to permu-tations of length 3 with the element 1 in the middle position, that is, the two permutations 213and 312.For the remaining permutations we break a permutation at the position where the element 1occurs. We obtain two smaller permutations σ and τ of lengths k , respectively, r , where k + r = n − (cid:0) n − k (cid:1) ways between these two permutations. This is encoded bymultiplication of exponential generating functions. Finally, the integral shifts the coefficient from w n − / ( n − z n /n !.We continue to describe the terms. The third term corresponds to 2 ≤ k, r , that is, at least twoelements precede the element 1 and at least two elements follow the element 1. Note that τ mustbegin with a descent to avoid creating a double ascent. Also when σ ends with a descent, we createa double descent when concatenating σ with the element 1. This explains the factor 2 in front ofthe term F x,b .The fourth term corresponds to k = 0 and r ≥
2. The Kronecker delta states that the per-mutation starts with an ascent. The fifth term corresponds to k = 1 and r ≥
2, in which thepermutation starts with a consecutive descent and ascent. Similarly, the sixth and seventh termscorrespond to the two cases r = 0 and k ≥
2, respectively, r = 1 and k ≥ Theorem 6.4.
The generating functions F x,y ( z ) and F ( z ) are given by F a , a ( z ) = 11 − z · (cid:0) e z − · e − z (cid:1) − · z,F a , b ( z ) = 11 − z · (cid:0) − · e − z (cid:1) + 1 − z,F b , b ( z ) = 11 − z · e − z − ,F ( z ) = 11 − z · (cid:0) e z − e − z (cid:1) . Proof.
Proposition 6.3 can be viewed as a recursion for the coefficient α n ( x, y ). Hence the equationin this proposition has a unique solution and it is enough to verify the theorem by showing thatthe proposed generating functions satisfy equation (6.1).Finally, the generating function F ( z ) is obtained by adding the four generating functions F a , a ( z ), F a , b ( z ), F b , a ( z ) and F b , b ( z ).Since e − z / (1 − z ) is the generating function for the number of derangements, we obtain Corollary 6.5.
For n ≥ , the number of derangements on n elements, D n , is given by α n ( b , b ) ,that is, D n = X π bb ( π ) , where the sum is over all permutations π on n elements with no double ascents and starting andending with a descent.
13s a corollary to Theorem 6.4 we have the following recursions:
Corollary 6.6.
Recursions for the sequences α n ( a , a ) , α n ( a , b ) , α n ( b , b ) and α n are given by,where n ≥ , α n ( a , a ) = n · α n − ( a , a ) + 1 + 4 · ( − n ,α n ( a , b ) = n · α n − ( a , b ) − · ( − n ,α n ( b , b ) = n · α n − ( b , b ) + ( − n ,α n = n · α n − + 1 + ( − n . Using the generating functions in Theorem 6.4 we now obtain that the error terms are thesmallest possible. We express the result as explicit expressions using the nearest integer function,which we denote by ⌊ x ⌉ . Theorem 6.7.
The quantities α n ( a , a ) , α n ( a , b ) , α n ( b , b ) and α n are given by the explicit expres-sions α n ( a , a ) = ⌊ ( e − /e ) · n ! ⌉ for n ≥ ,α n ( a , b ) = ⌊ (1 − /e ) · n ! ⌉ for n ≥ ,α n ( b , b ) = ⌊ /e · n ! ⌉ for n ≥ ,α n = ⌊ ( e − /e ) · n ! ⌉ for n ≥ . Proof.
The third equality is classical. We show the first equality. The coefficient of z n /n ! in thegenerating function F a , a ( z ), for n ≥
2, is given by α n ( a , a ) = n ! · n X k =0 k − · k + 4 · ( − k k ! . Hence the difference n ! · ( e − /e ) − α n ( a , a ) = n ! · X k ≥ n +1 k + 4 · ( − k k ! , is bounded in absolute value by n ! · X k ≥ n +1 k ! = 5 n + 1 + 5( n + 1) · ( n + 2) + · · · . Note that this is a decreasing function in n . For n = 10 this function dips below 1 /
2, showing thefirst equality for n ≥
10. The two cases n = 8 , Are there other operators of the form (2.1) which only have a finite number of non-zero eigenvalues?Furthermore, if the associated sequences are integer sequences would the corresponding error termbe the smallest possible, as in Theorem 6.7?The operators of the form (2.1) have so far yielded four types of behavior:14i) The operator has an infinite number of eigenvalues and the asymptotic expansion converges.An example of this is alternating permutations. See [3, Example 1.11] and [4]. Anotherexample is { , , } -avoiding permutations. See [3, Section 7].(ii) The operator has an infinite number of eigenvalues and the asymptotic expansion does not givean expression that converges. This occurs with 123-avoiding permutations and 213-avoidingpermutations. See [3, Sections 5 and 6].(iii) The operator has a finite, but positive, number of non-zero eigenvalues. See Section 6. Forinstance, is there such an operator with exactly two non-zero eigenvalues? What behaviordoes the error term of the asymptotic expansion have? Are there other examples with thesmallest possible error term?(iv) The operator has no non-zero eigenvalues. Here the behavior can vary a lot. Compare ba -avoiding permutations in Example 3.2 with { , } -avoiding permutations in [2]. Alsosee [3, Example 3.9].The two equations (5.1) and (5.2) in Section 5 have an interesting pattern in their roots.Consider the two equations in terms of the variable z = 1 /λ . Then the roots lie on the realaxis and close to a vertical line in the complex plane. Is there an explanation for this behavior?Switching back to the variable λ it says that the roots lie on the real axis and close to a circle inthe complex plane.Baxter, Nakamura and Zeilberger [1] have developed efficient methods to compute the num-ber of permutations avoiding certain patterns. Their methods use umbral techniques and havebeen implemented in Maple. Their techniques can be extended to compute the weighted problemintroduced in this paper.In Section 6 we obtained that the number of derangements D n is given by the sum over allpermutations with no double ascents, where each term is 2 to the power of the number of doubledescents. It is natural to ask for a bijective proof of this fact. In fact, Muldoon Brown andReaddy [6] gave essentially such a bijection.A descent run I in a permutation π = π π · · · π n is an interval I = [ i, j ] such that π i > π i +1 > · · · > π j . Observe that we do not require the interval to be maximal.Given a permutation π on n elements, write the interval [1 , n ] as a disjoint union S kr =1 I r ofdescent runs of the permutation π . A Muldoon–Readdy barred permutation σ from π is obtained bythe following procedure. For each descent run I r = [ i, j ] pick an element h r in the half-open interval( i, j ] and bar the elements π h r through π j . Observe that for a descent run I r of cardinality c thereare c − h r . Hence to obtain Muldoon–Readdy barred permutationeach descent run needs to have cardinality at least 2.Finally given a permutation π , how many Muldoon–Readdy barred permutations can be ob-tained from it? If the permutation π has a double ascent there is a maximal descent run of size 1and hence there is no way to obtain a Muldoon–Readdy permutation. Partition the interval [1 , n ]into maximal descent runs of π , that is, [1 , n ] = S s J s . Each maximal descent run J s can be furtherpartitioned into descent runs and be barred. Let the maximal descent run J s have cardinality c .Then the number of ways of partitioning J s and barring each descent run is X ( c ,c ,...,c k ) ( c − · ( c − · · · ( c k − , c . By the basic generating functionargument 1 / (1 − x / (1 − x ) ) = 1 + x / (1 − x ), the above sum is 2 c − . Finally, note that c − J r . Hence there are 2 to the number of doubledescents in π ways to obtain a Muldoon–Readdy barred permutation from the permutation π .Finally, Muldoon Brown and Readdy give a bijection between derangements and Muldoon–Readdy barred permutations. See Theorem 6.4 in [6]. This gives a bijective proof of Corollary 6.5. Acknowledgments
The authors thank Margaret Readdy and the two referees for their comments on an earlier draftof this paper. The authors are partially funded by the National Science Foundation grant DMS-0902063. The first author also thanks the Institute for Advanced Study and is also partiallysupported by the National Science Foundation grants DMS-0835373 and CCF-0832797.
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R. Ehrenborg and J. Jung, Department of Mathematics, University of Kentucky, Lexington, KY40506-0027, { jrge , jjung } @[email protected]