Combinatorial objects enumerated by q-bessel functions

Vol. 34 (1994) REPORTS ON MATHEMATICAL PHYSICS No. 1

COMBINATORIAL OBJECTS ENUMERATED BY q-BESSEL FUNCTIONS

J. M. FI~DOU

Laboratoire Bordelais de Recherche en Informatique, Universite Bordeaux I, 33405 Talence, France

(Received December 10, 1993)

We describe in this paper different combinatorial objects enumerated by q- Bessel functions : skew Ferrers diagrams according to width and area, simple braids according to number of threads and exponent and pairs of permutations according to common descents and inversions. These enumerations allow us to introduce different technics of counting and q-counting.

1. Introduction

The main tool in enumerative combinatorics is with no doubt the notion of generating function. The generating function of a given sequence of integers (an&v is defined by

f(x) = C adi?. nEN

One considers generating functions of several variables for multi-indexed sequences. The main reason for using generating functions is that recursions on sequences become functional equations, such as algebraic or differential equations. Usual tools of analysis are then particulary useful and powerful in deriving closed formulas or asymptotic expan- sions.

There are many ways of computing generating functions. In particular, when the generating function is algebraic, a nice way to prove its algebricity is to use the “DSV” or Schiitzenberger methodology. It consists in constructing a bijection between the objects and the words of an algebraic language such that the parameter of the enumeration is the length of the corresponding word. The generating function is the solution of a functional equation obtained from a non ambiguous grammar of the langage. A survey of this method is given in [29].

In fact if one analyses the reasons why this method works, one can see that the crucial point is not the bijection in itself, but the recursive definition of the objects generated

PI

58 J. M. FtiDOU

by the grammar of the langage. This led to the idea to eliminate the words in the “DSV” methodology and to work directly on the objects. We define what we call objects grammars [13] in order to explain equations such as the one in figure 2. We deduce from it functional equations satisfied by the generating function. The enumeration of the skew Ferrers diagrams [ll] developed in paragraph 2 is a representative example of this method.

Another way of deriving generating functions is to use inversion formulas such as Mobius inversion formulas. A very nice and powerful tool is the theory of heap of pieces, developed by X.G. Viennot [30], which is a visualization of the theory of partial commutative monoIds (also called trace mono’ids) introduced by Cartier and Foata [8]. The most complete study of trace mondids is the book of Diekert [14] ; a survey can be found in [26]. The theory of heaps of pieces allows us to give another combinatorial explanation for the generating function of skew Ferrers diagrams using simple braids.

Finally, we introduce a Mobius inversion theorem for the generating function of words in the free monoi’d according to descents. This theorem generalizes the well known

inversion formula in the free monoi’d and implies the inversion theorem in the monoid of trace when Dieckert condition holds [15]. We deduce the generating function of couples of permutations according to the number of common descents and inversions, which is a q-analog of a result of Carlitz, Scoville and Vaughan [7].

Here we deal with families of combinatorial objects whose generating functions are related to Bessel functions (See [31] for a complete study), and more precisely to basic Bessel or q-Bessel functions [21]. Bessel functions are usually defined by,

but we will consider here a combinatorial definition of Bessel functions, obtained from the classical one by a simple change of variables in which n is integer. We will denote in the same way,

There are at least three ways were introduced by Jackson [21] respectively,

J;qs; q) = c (-l)nZn+v n>O (dn(&+u -

of defining q-analogs of such functions. Two of them and were recently studied by Ismail [20], ‘which are

and n. n(n+vjZn+v

JL2)(? Q) = c (-l;qp (q) 7

QO n n+v

COMBINATORIAL OBJECTS ENUMERATED BY q-BESSEL FUNCTIONS 59

where as usual, (q)n = (1 - q)(l - q2) . . . (1 - qn). A third q-analog appeared recently,

&3)(x; q) = c (-~)“qvwx~+~

TL>O (q)n(q)n+v ’

in the enumeration of skew Ferrers diagrams [ll], and other kinds of polyominoes or animals [9], [24], [25].

2. Skew Ferrers diagrams and polyominoes

In this paragraph, we first introduce skew Ferrers diagrams and polyominoes and then illustrate objects grammars on the example of skew Ferrers diagrams. This allows us to obtain their generating function according to width and area.

2.1. Definitions

Ferrers diagrams, related to the well-known partitions of an integer, have been extensively studied. See for instance Andrews’ book [l]. A partition of an integer n is a decreasing sequence of integers, x = (ni, 122,. . . , nk), such that nr + n2 + . . . + nk = n. The geometric figure formed by the Ic rows having respectively nl, n2,. . . , nk cells is called the Ferrers diagram associated with the partition A of n. The difference between two Ferrers diagrams is called a skew Ferrers diagram. Thus a skew Ferrers diagram is defined by two increasing sequences of integers, (ni, 122, . . . , nk), and (pr,ps, . . . , pk) such that, for every 1 5 i 5 Ic, ni 5 pi (see figure 1).

Unit squares with vertices at integer coordinate points in the Cartesian plane are called cells. A polyomino is a finite connected union of cells such that the interior is also connected. Polyominoes are defined up to translation. Counting polyominoes according to the area or perimeter is a major unsolved problem in combinatorics. See for review [27], [28]. The problem is also well known in statistical physics. Usually, physicists consider animals instead of polyominoes, equivalent objects obtained by taking the center of each elementary cell. A column (resp. a row) is the intersection of the polyomino with an infinite vertical (resp. horizontal) unit strip. A polyomino is said to be convex when all its columns and rows are connected. Recently, convex polyominoes have been enumerated according to the width, height and area [9].

A parallelogram polyomino is a convex polyomino bordered by two nonintersecting paths having only North and East steps (see figure 1). Equivalently, connected skew Ferrers diagrams having no cut point are parallelogram polyominoes. For example, the skew Ferrers diagram shown in figure 1 is defined by the two sequences (5,4,4,2) and (1,l). It is also a parallelogram polyomino having perimeter 18 and area 13. Parallelogram polyominoes are well known in combinatorics . The number of such polyominoes having

perimeter 2n + 2 is the well-known Catalan number C, = & 2n

( > 7L . Parallelogram

polyominoes have been extensively studied first by Klarner and Rivest [22] according to

60 J. M. FI?DOU

Figure 1: A Skew Ferrers diagrams

area, by Delest and Fedou [ll] according to area and width, then by Brak and Guttman [4] according to area and site perimeter, and by Bousquet-Melou and Viennot [lo] according to width, height and area. More recently, Dubernard and Dutour [12] generalized their results by adding the parameter corners.

2.2. Enumeration of skew Ferrers diagrams

In this paragraph, we describe a method for deriving a functional equation satisfied by the generating function. This method, derived from Schiitzenberger methodology, is called object grammars method. It consists in defining recursively the objects we want to enumerate in a non-ambiguous way by applying recursively a finite number of object operations to a finite numbers of terminal objects. Functional equations are obtained by studying the effects of the operations on the statistics we want to compute. Let P denote the set of skew Ferrers diagrams. Let us define the associated objects operations,

l 91 : P -+ P defined by

“add a cell to the left of the south-west cell” ,

l cp2 : P + P defined by

“add a cell to the bottom of each column”,

l ?) : P x P + P defined by

“add a cell in each column of the first one and glue the south-west corner of the other one to the North-East of the first one”.

These three operations, together with the terminal object unit cell define a grammar object which generates P. It means that elements of P can be constructed unambiguously from unit cells by succesively applying the operations cpl, (~2 and +. Indeed, it is clear


0 + 0 Figure 2: A recursive definition of Skew Ferrers diagrams

that recursive application of these operations to elementary cells generates skew Ferrers diagrams. Conversely, given a skew Ferrers diagrams, it is possible to describe it by applying the operations : just pick out a cell in the bottom of each column until the remaining object is still a skew Ferrers diagram.

The visual equation showed figure 2 allows us to obtain the functional equation for the generating function of skew Ferrers diagrams according to width, height and area. Indeed, these statistics can be easily computed by reading the effect of the operations on an object. More precisely, denoting by w(P), h(P) and u(P) the width, height and area of a parallelogram p, and w(p) = zw(p)Yh(*)qa(p), we have,

l V(cpZ(P)) =YwdPMP),

so that the generating function f(x, y, q) satifies the q-equation

f(x, Y, 4) = xyq+ zqf(x, Y, Q) + Yf(X% Y, 4) + f(x, Y, q)f(xq, Y, 4.

THEOREM 1 The generating function for the skew Ferrer-s diairams according to width and area is,

Jj3)(4x. C?‘>

J$yx-‘q) . 7

I suffices to verify that the expected solution satisfies the above equation for y = 1. Details are given in [ll].

If one expands the function f (x, 1, q) in 2,

62 J. M. FEDOU

i-l i if1 i+2 i-l

1x1 I i-l i i+l i+2 i-l

Xi

i i+l i+2

i i+l i+2

Xi1

Figure 3: Generators of braids group

it appears that fn(q) is a polynomial in q, with positive unimodal coefficients. So a natural question is how to interpret combinatorialy these polynomials. This has been answered in [16] using simple braids.

3. Basic Bessel functions and simple braids

3.1. Braid group

The braid group B, has been introduced by Artin [2]. A recent survey is given in [3]. From a group theory point of view, the braid group B,is generated by x1, x2,. . . , x,+1 satisfying the relations

(i) for i, 1 5 i 5 ?I - 2, XiXi+lXi = Xi+lXiXi+l,

(ii) for i,j E {1,2.. . ,TZ - 1) such that ]i - j] > 2, ~i~j = xjxi.

Elements of the braid group are usualy visualized by threads having origins and extremities labelled from 1 to n. The generators of braid group are showed fig 3. Product of two braids is obtained by joining the extremities of the firs one to origins of the second one. Thus relation (ii) may be visualized as in figure 4 and corresponds to the usual notion of homotopy for knots. A classical statistic for braids is the exponent. For a braid t defined by a product of generators w, the exponent is defined by

exp(t) = C 1~1,~ (1;; ) - @Y$.

This number is independent of the chosen representative w. The trace of a braid t is z?(t) = q-w

3.2. Simple braids

We are dealing here with a subset of the braid group whose elements are called simple braids. A simple braid is a braid for which the threads are ordered by the relation “over”,


Figure 5: Simple and non simple braids.

that is which is obtained by piling threads one over the other. For instance the braid T of figure 5 is simple because the treads (12) and (24) are over (33) which is over (41). Such relations are not possible for the braid T’, so T’ is not a simple braid. Note that 3, contains a finite number of simple braids and that the product of simple braids is generally not simple. Let SS, be the set of simple braids of B,. Let BSA be the set of simple braids with a unique maximal thread having its extremity labelled by 1. We are now able to formulate the main theorem relating simple braids to q-Bessel functions [16].

THEOREM 2 The polynomials

a,(q) = q(T) C tr(t) CBS,

and

satisfy the relation

b,(q) = q(B) c tr(t)

1

J~s)(2’ q) > = Can(q)&

TX>0 n

64 J. M. Fl?DOU

The proof of this theorem is detailed in [16], and is based on the theory of heap of pieces. We will now give a short introduction to this theory and outline the proof.

3.3. Heaps of pieces

The theory of heap of pieces, due to Viennot [30] is a geometrical version of the theory of partial commutative mondids or traces monoi’ds, introduced by P. Cartier and D. Foata [8]. Let us recall the basic definitions and results of [30]. Let X be a set of basic pieces, and R a reflexive symmetric relation on X. Two basic pieces p and p’ in X are said to be concurrent if (p R p’) and independant if not. A piece is a pair (basic piece, level) where level is a non negative integer. A heap of pieces is a set of pieces satisfying the following conditions,

(i) two concurent pieces have different levels,

(ii) for every piece P having a nonzero level p, there exists at least a piece P’ having level p-l and concurrent to P.

The set of the heap of pieces is denoted by P(X,R), or simply P when there is no possible confusion. A piece p is said to be maximal when any piece in concurrence with p stands at a lower level. The set of maximal pieces of a heap E is denoted by p(E). A heap of piece is said to be trivial when any two pieces are mutually independant. The set of trivial heaps of pieces is denoted by T. A valuation is a map from the set of pieces into a polynomial ring such that the valuation of the empty heap is 1. The valuation of a heap of pieces is the product of pieces of the heap. The generating function of a set S of heaps of pieces is the sum of the valuations of the heaps of S. One of the main results of the theory of heaps of pieces is to provide Mobius inversion theorems for generating functions of heaps of pieces:

THEOREM 3 The generating function of heap of pieces is

c v(E) = c (&(t) .

EEP tET

THEOREM 4 The generating function of heap of pieces having their maximal pieces in a certain set M of basic pieces is

c (-lpw(t)

COMBINATORIAL OBJECTS ENUMERATED BY *-BESSEL FUNCTIONS 65

4 (1) A;(4) = A; G(5) n I n n

0 Ad‘4 Ad% A3(5) A4(6)

Figure 6: A trivial heap of sticks.

3.4. Heaps of sticks

Theorem 3 and 4 allow us to describe

1

@)(2. q) and

7

using heaps of sticks. Let D and D’ be two parallel lines in the 3-dimensional space. A stick is a half-segments [A(a),A’(a’)] of p oin s such that A (resp. A’) is point of D t (resp.D’) with integer coordinates a (resp.a’). Let us define the concurrence relation R by [A(a), A’(a’)] R [B(b),B’(b’)] iff (a 5 b and a’ > b’) or (a 2 b and a’ < b’). Applicaion of theorem 3 allows us to deduce that the function

1

J($q-c; q)

is the generating function of heap of sticks. In the same Way,

/3)(x* q)

$+cY q) 7

is the generating function of heaps of sticks having a unique maximal piece [A,A’] for which A has coordinate equal to zero. It suffices to study trivial heaps of sticks as showed figure 6 and to apply theorems 3 and 4.

3.5. Heaps of sticks and simple braids

The proof of theorem 3 is deduced from a family of bijections (Pi between heaps of n sticks and triples (simple braid, increasing sequence, strictly increasing sequence)

66 J. M. FEDOU

which preserves the weights. The bijections are recursively defined by applying a “shift- insertion” principle. Let H be a heap of n sticks. Suppose that [A(a), A’(a’)] is its left- most maximal stick and denote by H’ the heap obtained from H by removing [A,A’]. Sup- pose also that the image of H’ by the bijection (~~-1 is cp,_i(H) = (‘u., (bi)yzl, (bi)yz;). Set b, = b; = +co. There are unique integers i,j such that,

bl 5 b2 2 . . . 2 bi-1 5 a < bi 2 . . . 5 b, and bi < bh < . . . < bi_, < a’ 5 bi < . . . < b;.

Define the sequences

ak=bkifk<i,ai=a and ak=bk-r-lifIc>i,

and ak=bkifk<j,aj=a and ak=bk_i+lifIc>j.

Let t = (xi.. . ~~)-~u(xj.. . z,) be the simple braid obtained from u by first shifting origines greater than i and extremities greater than j, then adding the thread (i,j) on the top. Thus the bijection (Pa is defined by,

The proof of theorem 2 is obtained by studying the properties of this bijection.

Details of paragraphs 3.4 and 3.5 are given in [16]

4. Basic Bessel functions and couples of permutations

4.1. Eulerian generating functions

First note that simple braids are in bijection with couples of permutations without descents. It suffices to “read” simple braids from up to down and from left to right. Such couples of permutations have been studied by Carlitz, Scoville and Vaughan in ref. [7], where they compute the “Bessel” generating function of couples of permutations according to their size and number of common descents. More precisely, if B,,I, denotes the number of couple of permutations having k common descents,

This formula can be completly q-analogised . Full details are given in [17] and the proof is a q-analog of Carlitz’s one. The second remark is that such inversion formula, involving alternate sums of (1 - y)“- ‘9 is also present in the well known Euler polynomials and q-Euler polynomials ([5] and [6]). Therefore we tried to “explain” Eulerian generating functions as

1

1 + En21 (-lpX,(l- y)n-12n


in a similar way that heaps of pieces “explain”

1

The key of the answer is the notion of descent in the free mono’id.

4.2. Descents in a word

Let an alphabet X be given, lexicographically ordered, and a symmetric antireflexive relation 8. We say that a word w = x1 . . . x~x~+~ . . . x, of X” has a edescent (or simply a descent when there is no possible confusion) at position i when

(i) xiexi+l,

(ii) xi > xi+1 (in the sense of the lexicographic order).

We denote by a >>e b (resp. a <<e b ) when aeb and a > b (resp.(a < b)). Let 16 = {Xi.. .x,, : n E IV, x1 23 x2 >O . . . >>e x,-l >>e x,,xi E X}. For a word w in X* ODes(w) denotes the set of positions of the Bdescents of w, Odes the number of its descents. Again, we will use a << b, I, Des(w) and des(w) when there is no possible confusion. Notation. We denote by I+ the set of the non-empty words 21x2.. .x, XlX2...X, such that x1 >> 22 >> . . . >> x,

THEOREM 5 The following equality holds in the free monoi’d

c yw4 w = 1 + t:

WEX’ tcl+ (-l;l4 (1 - y)l+-1 t *

EXAMPLE. Let X = {a, b, c} with the usual lexical order a < b < c. Consider the relation 0 = {(a, b), (b, a)} Then the 8 descents of a word of X* are the factors ba. Moreover we have I+ = {a, b, c, ba} and theorem 5 can be written

y yI{w~=~b~~)lW = 1

1 - (a + b + c) + (1 - y)ba ’ -- WEX’

REMARK. Theorem 4 can be deduced by applying theorem 5 and setting y = 0 only when the relation >> is transitive which is the condition of Dieckert [14].

4.3. Application to permutations

A descent of a permution s is an integer i such that s(i) > s(i + 1). An inversion is a couple of integers i < j such that s(i) > s(j). Permutations are classically enumerated according to their descents and inversions by q-Eulerian polynomials [6],

68 J. M. FlkDOU

Consider the alphabet N ordered by the usual relation <, and the trivial relation iej when i # j. Let us associate to each letter i the weight qi. Theorem 5 becomes,

+

yl{ki>ki-l)l : qlcl+...+kn ). : xn 7X20 hr.. ,kn

1 1

= ’ - Cn>l c1 - Y)“-’ (ck,,...,,, q”l+‘.‘+“-) : xn = _ 1 _ .j& (1 _ y,n-‘f$: xn 7%

Moreover, applying an “insertion-shift” principle, it is possible to associate to each word of N* a couple {permutation, increasing sequence} and deduce from the above equality,

which is another definition of q-Eulerian polynomials [6].

4.4. Application to couples of permutations

Applying an analogue technique, we consider the alphabet N x N, usually lexicographically ordered, and the relation (a, b) 0 (a’, b’) when (u > a’ and b 5 b’) or (U < a’ and b > b’). Applying the weight qf : qi to each letter (a, b), one obtains by theorem 5 the generating function

1

1 - &,I (1 - Y)n-‘$&

Again, an “insertion-shift” principle allows to associate to each word a triple {couple of permutations, strictly increasing sequence, increasing sequence} preserving the weights and descents. Let

%(91,92r Y) = c Ycin”(“‘%, des(a)qp-wP)

>

(%P)Es:

where cinv(a,b) denotes the number of common descents of the permutations a and b. This bijection allows us to prove a new result, which is a (ql,q2)-analog of [7],

THEOREM 6 The Laurent polynomials B,(q, y) satisfy the relation,

. COMBINATORIAL OBJECTS ENUMERATED BY n-BESSEL FUNCTIONS 69

Details and generalizations will be given in [18].

Conclusions. We presented in this paper some technics for computing ordinary generating functions. Considerable work has been done in this area. Exponential generating functions &o a,% and q-exponential functions, ‘&s a,& also have been exten-

sively studied,But very little have been done in the field of Bessel generating functions

c ,+s a,$ and q-Bessel generating functions En,,, a,& and we are sure that there

is still much to be done in this area. -

REFERENCES

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[l?] Dubernard, J. P., Dutour, I.: Enumeration of directed convex polyominoes, Rapport de Recherche 98-93, LaBRI, Universite de Bordeaux I.

[13] Dutour, I., Fedou, J. M.: Grammaires d’objets, in preparation.

[14] Diekert, V.: Combinatorics on Traces, Lecture Notes in Comp. Sci. 454 (1987).

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[16] Fedou, J. M.: Fonctions de Bessel, Empilements et Tresses, &tes du 4e colloque “Series formelles et combinatoire algebrique” , 189-202, U_niversitC du Quebec a Mon- treal 1992.

[17] Fedou, J. M., Rawlings, D.: Statistics on pair of permutations, to appear in Discrete Mathematics.

[18] Fedou, J. M., Rawlings, D.: Descents in Words, in preparation.

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[23] Mazurkiewicz, A.: Lect. Notes in Comp. Sci. 255 (1987) 279-324.

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