Báo cáo toán học: "Some Aspects of Hankel Matrices in Coding Theory and Combinatorics Ulrich Tamm" ppt

MR Subject Classifications: primary 05A15, secondary 15A15, 94B35 1≤i≤j≤k i+j−1+2n i+j−1 can be shown to hold for Hankel matrices whose entries are successive middle binomial coefficient

Some Aspects of Hankel Matrices in Coding Theory

Submitted: December 8, 2000; Accepted: May 26, 2001.

MR Subject Classifications: primary 05A15, secondary 15A15, 94B35

1≤i≤j≤k i+j−1+2n i+j−1 can be shown

to hold for Hankel matrices whose entries are successive middle binomial coefficients 2m+1

m

.Generalizing the Catalan numbers in a different direction, it can be shown that determinants ofHankel matrices consisting of numbers 1

3m+1 3m+1 m

yield an alternate expression of two Mills –Robbins – Rumsey determinants important in the enumeration of plane partitions and alternat-ing sign matrices Hankel matrices with determinant 1 were studied by Aigner in the definition

of Catalan – like numbers The well - known relation of Hankel matrices to orthogonal mials further yields a combinatorial application of the famous Berlekamp – Massey algorithm inCoding Theory, which can be applied in order to calculate the coefficients in the three – termrecurrence of the family of orthogonal polynomials related to the sequence of Hankel matrices

For a sequence c0, c1, c2, of real numbers we also consider the collection of Hankel

So the parameter n denotes the size of the matrix and the 2n − 1 successive elements

c k , c k+1 , , c k+2n−2 occur in the diagonals of the Hankel matrix

We shall further denote the determinant of a Hankel matrix (1.2) by

d (k) n = det(A (k) n ) (1.3)

Hankel matrices have important applications, for instance, in the theory of moments,and in Pad´e approximation In Coding Theory, they occur in the Berlekamp - Masseyalgorithm for the decoding of BCH - codes Their connection to orthogonal polynomialsoften yields useful applications in Combinatorics: as shown by Viennot [76] Hankel deter-minants enumerate certain families of weighted paths, Catalan – like numbers as defined

by Aigner [2] via Hankel determinants often yield sequences important in combinatorialenumeration, and as a recent application, they turned out to be an important tool in theproof of the refined alternating sign matrix conjecture

The framework for studying combinatorial applications of Hankel matrices and furtheraspects of orthogonal polynomials was set up by Viennot [76] Of special interest aredeterminants of Hankel matrices consisting of Catalan numbers 2m+11 2m+1 m

de-of Young tableaux, matchings, etc

They studied (1.4) as a companion formula forQ

1≤i≤j≤k i+j−1+c i+j−1 , which for integer c was

shown by Gordon (cf [67]) to be the generating function for certain Young tableaux

For even c = 2n this latter formula also can be expressed as a Hankel determinant formed

of successive binomial coefficients 2m+1 m

We are going to derive the identities (1.4) and (1.5) simultaneously in the next section.Our main interest, however, concerns a further generalization of the Catalan numbers andtheir combinatorial interpretations.

In Section III we shall study Hankel matrices whose entries are defined as generalized

2 4j−1 2j
(1.6)

These numbers are of special interest, since they coincide with two Mills – Robbins – sey determinants, which occur in the enumeration of cyclically symmetric plane partitionsand alternating sign matrices which are invariant under a reflection about a vertical axis.The relation between Hankel matrices and alternating sign matrices will be discussed inSection IV

Rum-Let us recall some properties of Hankel matrices Of special importance is the equation

In Section V we further shall discuss the Berlekamp – Massey algorithm for the decoding

of BCH–codes, where Hankel matrices of syndromes resulting after the transmission of a

code word over a noisy channel have to be studied Via the matrix L n defined by (1.11)

it will be shown that the Berlekamp – Massey algorithm applied to Hankel matriceswith real entries can be used to compute the coefficients in the corresponding orthogonalpolynomials and the three – term recurrence defining these polynomials

Several methods to find Hankel determinants are presented in [61] We shall mainlyconcentrate on their occurrence in the theory of continued fractions and orthogonal poly-

nomials If not mentioned otherwise, we shall always assume that all Hankel matrices A n

under consideration are nonsingular

Hankel matrices come into play when the power series

is expressed as a continued fraction If the Hankel determinants d(0)n and d(1)n are different

from 0 for all n the so–called S–fraction expansion of 1 − xF (x) has the form

For the notion of S– and J– fraction (S stands for Stieltjes, J for Jacobi) we refer to the

standard books by Perron [55] and Wall [78] We follow here mainly the (q n , e n)–notation

of Rutishauser [65]

For many purposes it is more convenient to consider the variable 1

x in (1.13) and studypower series of the form

Hankel determinants occur in Pad´e approximation and the determination of the ues of a matrix using their Schwarz constants, cf [65] Especially, they have been studied

eigenval-by Stieltjes in the theory of moments ([70], [71]) He stated the problem to find out if a

measure µ exists such that

Stieltjes could show that such a measure exists if the determinants of the Hankel

ma-trices A(0)n and A(1)n are positive for all n Indeed, then (1.9) results from the quality of

the approximation to (1.16) by quotients of polynomials p t j (x)

j (x) where t j (x) are just the

polynomials (1.8) Hence they obey the three – term recurrence

t j (x) = (x − α j )t j−1 (x) − β j−1 · t j−2 (x), t0(x) = 1, t1(x) = x − α1, (1.19)

where

α1 = q1, and α j+1 = q j+1 + e j , β j = q j e j for j ≥ 1 (1.20)

In case that we consider Hankel matrices of the form (1.2) and hence the corresponding

power series c k + c k+1 x + c k+2 x2+ , we introduce a superscript (k) to the parameters

II Hankel Matrices and Chebyshev Polynomials

Let us illustrate the methods introduced by computing determinants of Hankel matriceswhose entries are successive Catalan numbers In several recent papers (e.g [2], [47],[54], [62]) these determinants have been studied under various aspects and formulae weregiven for special parameters Desainte–Catherine and Viennot in [24] provided the general

solution d (k) n =Q

1≤i≤j≤k−1 i+j+2n i+j for all n and k This was derived as a companion formula

(yielding a “90 % bijective proof” for tableaux whose columns consist of an even number

of elements and are bounded by height 2n) to Gordon’s result [36] in the proof of the

Bender – Knuth conjecture [8] Gordon proved that Q

1≤i≤j≤k c+i+j−1 i+j−1 is the generatingfunction for Young tableaux with entries from {1, , n} strictly increasing in rows and

not decreasing in columns consisting of ≤ c columns and largest part ≤ k Actually, this

follows from the more general formula in the Bender – Knuth conjecture by letting q → 1,

see also [67], p 265

By refining the methods of [24], Choi and Gouyou – Beauchamps [21] could also derive

Gordon’s formula for even c = 2n In the following proposition we shall apply a well

-known recursion for Hankel determinants allowing to see that in this case also Gordon’sformula can be expressed as a Hankel determinant, namely the matrices then consist ofconsecutive binomial coefficients of the form 2m+1 m

Simultaneously, this yields anotherproof of the result of Desainte – Catherine and Viennot, which was originally obtained byapplication of the quotient – difference algorithm [77]

This identity can for instance be found in the book by Polya and Szeg¨o [59], Ex 19, p

102 It is also an immediate consequence of Dodgson’s algorithm for the evaluation ofdeterminants (e.g [82])

We shall derive both results simultaneously The proof will proceed by induction on n+k.

It is well known, e.g [69], that for the Hankel matrices A (k) n with Catalan numbers as

entries it is d(0)n = d(1)n = 1 For the induction beginning it must also be verified that

d(2)n = n + 1 and that d(3)n = (n+1)(n+2)(2n+3)6 is the sum of squares, cf [47], which can also

be easily seen by application of recursion (2.3)

Furthermore, for the matrix A (k) n whose entries are the binomial coefficients 2k+1

k

, 2k+3

k+1

,

it was shown in [2] that d(0)n = 1 and d(1)n = 2n + 1 Application of (2.3) shows that

d(2)n = (n+1)(2n+1)(2n+3)3 , i e., the sum of squares of the odd positive integers

Also, it is easily seen by comparing successive quotients c k+1

c k that for n = 1 the product in

(2.1) yields the Catalan numbers and the product in (2.2) yields the binomial coefficients

2k+1

k+1

, cf also [24]

Now it remains to be verified that (2.1) and (2.2) hold for all n and k, which will be done

by checking recursion (2.3) The sum in (2.3) is of the form (with either d = 0 for (2.1)

or d = 1 for (2.2) and shifting k to k + 1 in (2.1))

1) As pointed out in the introduction, Desainte–Catherine and Viennot [24] derived

iden-tity (2.1) and recursion (2.3) simultaneously proves (2.2) The ideniden-tity det(A(0)n ) = 1,

when the c m’s are Catalan numbers or binomial coefficients 2m+1 m

can already be found

in [52], pp 435 – 436 d(1)n , d(2)n , and d(3)n for this case were already mentioned in the proof

of Theorem 2.1 The next determinant in this series is obtained via d(4)n

2) Formula (2.1) was also studied by Desainte–Catherine and Viennot [24] in the analysis

of disjoint paths in a bounded area of the integer lattice and perfect matchings in a

certain graph as a special Pfaffian An interpretation of the determinant d (k) n in (2.1) as

the number of k–tuples of disjoint positive lattice paths (see the next section) was used

to construct bijections to further combinatorial configurations Applications of (2.1) inPhysics have been discussed by Guttmann, Owczarek, and Viennot [40]

3) The central argument in the proof of Theorem 2.1 was the application of recursion(2.3) Let us demonstrate the use of this recursion with another example Aigner [3]

could show that the Bell numbers are the unique sequence (c m)m=0,1,2, such that

l=1 n(n − 1) · · · (n − l + 1) is the total number of permutations of n

things (for det(A(0)n ) and det(A(1)n ) see [27] and [23]) In [3] an approach via generating

functions was used in order to derive d(2)n = det(A(2)n ) in (2.4) Setting d(2)n = r n+1 ·Qn

k=0 k!

in (2.4), with (2.3) one obtains the recurrence r n+1 = (n + 1) · r n + 1, r2 = 5, which just characterizes the total number of permutations of n things, cf [63], p 16, and hence can derive det(A(2)n ) from det(A(0)n ) and det(A(1)n ) also this way

4) From the proof of Proposition 2.1 it is also clear thatQ

1≤i,j≤k i+j−d+2n i+j−d yields a sequence

of Hankel determinants d (k) n only for d = 0, 1, since otherwise recursion (2.3) is not fulfilled.

As pointed out, in [24] formula (2.1) was derived by application of the quotient – difference

algorithm, cf also [21] for a more general result The parameters q n (k) and e (k) n also can

be obtained from Proposition 2.1

Corollary 2.1: For the Catalan numbers the coefficients q (k) n and e (k) n in the continuedfractions expansion ofP∞

For the binomial coefficients 2m+1 m

the corresponding coefficients in the expansion of

Proof: (2.5) and (2.6) can be derived by application of the rhombic rule (1.21) and (1.22).They are also immediate from the previous Proposition 2.1 by application of (1.15), which

for k > 0 generalizes to the following formulae from [65], p 15, where the d (k) n ’s are Hankeldeterminants as (1.3)

α (k) n+1= 2− 2k(k − 1)

(2n + k + 2)(2n + k) , β

(k)

n = (2n + 2k − 1)(2n + 2k)(2n)(2n + 1) (2n + k − 1)(2n + k)2(2n + k + 1) . Proof: By (1.20), β n (k) = q n (k) · e (k) n as in the previous corollary and

α (k) n+1 = q n+1 (k) + e (k) n = (2n + 2k + 1)(2n + 2k + 2)((2n + k) + (2n)(2n + 1)(2n + k + 2)

(2n + k + 1)(2n + k + 2)(2n + k)

= 8n2+ 8nk + 8n + 2k + 4k2(2n + k + 2)(2n + k) = 2− 2k(k − 1)

(2n + k − 1)(2n + k)2(2n + k + 1) .

Proof: Again, β n (k) = q n (k) · e (k) n as in the previous corollary and

α (k) n+1 = q (k) n+1 + e (k) n = (2n + 2k + 2)(2n + 2k + 3)((2n + k) + (2n − 1)(2n)(2n + k + 2)

(2n + k)(2n + k + 1)(2n + k + 2)

= 8n2+ 8nk + 8n + 2k2+ 4k (2n + k + 2)(2n + k) = 2− 2k(k + 1)

(2n + k + 2)(2n + k) .

III Generalized Catalan Numbers And Hankel Determinants

For an integer p ≥ 2 we shall denote the numbers pm+11 pm+1 m

as generalized Catalan

numbers The Catalan numbers occur for p = 2 (The notion “generalized Catalan

numbers” as in [42] is not standard, for instance, in [39], pp 344 – 350 it is suggested todenote them “Fuss numbers”)

Their generating function

count the number ofpaths in the integer lattice Z×Z (with directed vertices from (i, j) to either (i, j + 1) or

to (i + 1, j)) from the origin (0, 0) to (m, (p − 1)m) which never go above the diagonal (p − 1)x = y Equivalently, they count the number of paths in Z×Z starting in the

origin (0, 0) and then first touching the boundary {(l + 1, (p − 1)l + 1) : l = 0, 1, 2, } in (m, (p − 1)m + 1) (cf e.g [75]).

Viennot [76] gave a combinatorial interpretation of Hankel determinants in terms of joint Dyck paths In case that the entries of the Hankel matrix are consecutive Catalannumbers this just yields an equivalent enumeration problem analyzed by Mays and Woj-ciechowski [47] The method of proof from [47] extends to Hankel matrices consisting ofgeneralized Catalan numbers as will be seen in the following proposition

dis-Proposition 3.1: If the c m ’s in (1.2) are generalized Catalan numbers, c m = pm+11 pm+1 m

,

p ≥ 2 a positive integer, then det(A (k) n ) is the number of n–tuples (γ0, , γ n−1) of vertex

– disjoint paths in the integer lattice Z×Z (with directed vertices from (i, j) to either (i, j + 1) or to (i + 1, j)) never crossing the diagonal (p − 1)x = y, where the path γ r isfrom (−r, −(p − 1)r) to (k + r, (p − 1)(k + r)).

Proof: The proof follows the same lines as the one in [32], which was carried out only for

the case p = 2 and is based on a result in [46] on disjoint path systems in directed graphs.

We follow here the presentation in [47]

Namely, letG be an acyclic directed graph and let A = {a0, , a n−1 }, B = {b0, , b n−1 }

be two sets of vertices in G of the same size n A disjoint path system in (G, A, B) is a

system of vertex disjoint paths (γ0, , γ n−1 ), where for every i = 0, , n − 1 the path

γ i leads from a i to b σ(i) for some permutation σ on {0, , n − 1}.

Now let p ij denote the number of paths leading from a i to b j in G, let p+ be the number

of disjoint path systems for which σ is an even permutation and let p − be the number

of disjoint path systems for which σ is an odd permutation Then det((p ij)i,j=0, ,n−1) =

Further let A = {a0, a n−1 } and B = {b0, b n−1 } be two sets disjoint to each other

and to V Then we connect A and B to G 0 by introducing directed edges as follows

a i → (−i, −(p − 1)i), (k + i, (p − 1)(k + i)) → b i , i = 0, , n − 1 (3.4)

Now denote byG 00the graph with vertex set V ∪ A ∪ B whose edges are those from G 0 and

the additional edges connecting A and B to G 0 as described in (3.4).

Observe that any permutation σ on {0, , n − 1} besides the identity would yield some

j and l with σ(j) > j and σ(l) < l But then the two paths γ j from a j to b σ(j) and γ l

from a l to b σ(l) must cross and hence share a vertex So the only permutation yielding

a disjoint path system for G 0 is the identity The number of paths p ij from a i to b j isthe generalized Catalan number p(k+i+j)+11 p(k+i+j)+1 (k+i+j)

So the matrix (p ij) is of Hankel

type as required and its determinant gives the number of n – tuples of disjoint paths as

Remarks:

1) The use of determinants in the enumeration of disjoint path systems is well known,e.g [31] In a similar way as in Proposition 3.1 we can derive an analogous result for thenumber of tuples of vertex – disjoint lattice paths, with the difference that the paths now

are not allowed to touch the diagonal (p − 1)x = y before they terminate in (m, (p − 1)m) Since the number of such paths from (0, 0) to (m, (p−1)m) is pm+p−11 pm+p−1 m+1

di-by assigning weights to the steps in such a path, which are obtained from the coefficients

in the three–term recurrence of the orthogonal polynomials ([76], cf also [26]) In the

case that all coefficients α j are 0, a Dyck path arises with vertical steps having all weight

1 and horizontal steps having weight β j for some j For the Catalan numbers as entries

in the Hankel matrix all β j’s are 1, since the Chebyshev polynomials of second kind arise

So the total number of all such paths is counted Observe that Proposition 3.1 extendsthe path model for the Catalan numbers in another direction, namely the weights of thesingle steps are still all 1, but the paths now are not allowed to cross a different boundary

In order to evaluate the Hankel determinants we further need the following identity

Lemma 3.1: Let p ≥ 2 be an integer Then

of lattice paths (where possible steps are

from (i, j) to either (i, j + 1) or to (i + 1, j)) from (0, 0) to (m, (p − 1)m + 1) in a second way Namely each such path must go through at least one of the points (l, (p − 1)l + 1),

l = 0, 1, , m Now we divide the path into two subpaths, the first subpath leading from

the origin (0, 0) to the first point of the form (l, (p − 1)l + 1) and the second subpath from (l, (p − 1)l + 1) to (m, (p − 1)m + 1) Recall that there are pl+11 pl+1 l

possible choicesfor the first subpath and obviously there exist p(m−l) m−l

possibilities for the choice of the

Theorem 3.1: For m = 0, 1, 2 let denote c m = 3m+11 3m+1 m

and b m = 3m+21 3m+2 m+1

.Then

2 4j−1 2j
(3.6)

For quotients of such hypergeometric series the continued fractions expansion as in (1.14)

was found by Gauss (see [55], p 311 or [78], p 337) Namely for n = 1, 2, it is

e n= (α + n)(γ − β + n) (γ + 2n)(γ + 2n + 1) , q n =

(β + n)(γ − α + n) (γ + 2n − 1)(γ + 2n) . Now denoting by q n (D) and e (D) n the coefficients in the continued fractions expansion of

the power series D(x) = 1 − xC3(x)2 under consideration, then taking into account that

y = 274x we obtain with the parameters in (3.8) that

e (D) n = 3

2

(6n + 1)(3n + 2) (4n + 1)(4n + 3) , q

(D)

n = 32

(6n − 1)(3n + 1) (4n − 1)(4n + 1) . (3.9) The continued fractions expansion of 1 + xC3(x)2 differs from that of 1− xC3(x)2 only by

changing the sign of c0 in (1.14).

So, by application of (1.15) the identity (3.7) for the determinants d(0)n and d(1)n of Hankelmatrices with the numbers 3m+21 3m+2

(6n − 1)(3n + 1) (4n − 1)(4n + 1) =

2(6n)(6n − 1)(2n)(3n + 1) (4n + 1)(4n)2(4n − 1)

= (3n + 1)(6n)!(2n)!

(4n + 1)!(4n)! · 2 4n−1 2n

6n−2 2n

(6n + 4)(6n + 3)(6n + 2)(6n + 1)(2n + 1) 2(4n + 3)(4n + 2)2(4n + 1)(3n + 1)

=

6n+4 2n+2

where d(0)n−1 , d(1)n−1 , d(0)n , d(1)n , d(0)n+1 are the determinants for the Hankel matrices in (3.7)

In order to find the determinants for the Hankel matrices in (3.6) with generalized Catalannumbers 3m+11 3m+1 m

as entries, just recall that D(x) = 1 − xC3(x)2 = C1

is obtained by setting q (C)1 = 1, e (C) n = q n (D) for n ≥ 1 and q (C) n = e (D) n−1 for n ≥ 2. 

Problem: In the last section we were able to derive all Hankel determinants d (k) n with

Catalan numbers as entries So the case p = 2 for Hankel determinants (1.2) consisting of

numbers pm+11 pm+1 m

is completely settled For p = 3, the above theorem yields d(0)n and

d(1)n However the methods do not work in order to determine d (k) n for k ≥ 2 Also they

do not allow to find determinants of Hankel matrices consisting of generalized Catalan

numbers when p ≥ 4 What can be said about these cases?

Let us finally discuss the connection to the Mills – Robbins – Rumsey determinants