A continuous analogue of lattice path enumeration
aa r X i v : . [ m a t h . C O ] A ug A CONTINUOUS ANALOGUE OF LATTICE PATH ENUMERATION
TANAY WAKHARE † , CHRISTOPHE VIGNAT ‡ , QUANG-NHAT LE § , AND SINAI ROBINS ∗ Abstract.
Following the work of Cano and Díaz, we consider a continuous analog of lattice pathenumeration. This allows us to define a continuous version of any discrete object that counts certaintypes of lattice paths. As an example of this process, we define continuous versions of binomialand multinomial coefficients, and describe some identities and partial differential equations theysatisfy. Finally, we illustrate a general method to recover discrete combinatorial quantities from theircontinuous analogs. Introduction
Cano and Díaz [1, 2] have recently explored a novel method of obtaining continuous analoguesof discrete objects such as binomial coefficients and Catalan numbers. First, they realized thesediscrete quantities as the number of certain lattice paths . Next, they considered directed paths as continuous extensions of lattice paths and define moduli spaces of directed paths. Finally, theydeclared the volumes of these moduli spaces to be the continuous versions of the original discreteobjects.Here we extend some of Cano and Díaz’s work to higher dimensions, obtaining a partial differentialequation that the continuous multinomials satisfy, which generalizes the partial differential equationof Cano and Díaz from dimension to dimension n . Finally, we show how to use Todd operators todiscretize these moduli spaces, enabling us to retrieve the number of lattice paths from their associatedmoduli space.One motivation for extending the study of lattice paths to a natural continuous analogue is that it isdifficult to count discrete lattice paths satisfying various geometric constraints; it might be much easierto compute the volume of a related polytope (or union of polytopes) in some cases. This can in turngive us natural bounds for the number of lattice paths, and especially lattice paths under additionalgeometric constraints.We begin by describing the work of Cano and Díaz in some detail. Consider a collection of vectors W = { w , . . . , w N } in Z d which all lie on the same side of some fixed hyperplane containing theorigin. The vectors w i are called admissible directions . We define a lattice path as an ordered ( n + 1) -tuple of integer vectors ( , p , . . . , p n ) , with each p j ∈ Z d , and where(1.1) p k := p k − + λ k w c k , for some w c k ∈ W , and λ k ∈ Z ≥ . Intuitively, a lattice path is a finite path in Z d that follows (someof) the directions w , . . . , w N using integer steps. The classical example of lattice path counting isthe binomial coefficient (cid:0) ba (cid:1) , which counts the number of lattice paths in R from the origin to a point q := ( a, b − a ) ∈ Z ≥ , using the directions w := (1 , and w := (0 , . It is also desirable to give a − correspondence between each lattice path and the relevant λ k that define it, as is done in (1.2)below.We would like to explore the space of all paths from the origin to some fixed q ∈ R d , still usingthe admissible directions W , but now using real coefficients. Following Cano and Díaz, we define a directed path using the same set-up as in definition (1.1) above except for the important differencethat now each coefficient λ i is a non-negative real number. Consider the set of all directed paths fromthe origin to a fixed q ∈ R d , using the set of directions from the admissible directions W . That is, wedefine (1.2) P ( q , c ) := { ( λ , . . . , λ n ) ∈ R n ≥ | λ w c + · · · + λ n w c n = q } , for some w c , . . . , w c n ∈ W . By definition, P ( q , c ) is a polytope, which we call a path polytope .We call the collection of indices used here, namely c := ( c , . . . , c n ) , a pattern for the directed paths.It also follows easily from this definition that the path polytope P ( q , c ) ⊂ R n has dimension n − d .Most importantly, we can interpret the set of integer points in P ( q , c ) as the set of lattice paths , withpattern c , defined by (1.1). In other words, we define(1.3) L ( q , c ) := { P ( q , c ) ∩ Z d } , the set of integer points in P ( q , c ) , which is also the set of lattice paths (from to q ) that use thesubset w c , . . . , w c n of the admissible directions W .We next describe the moduli space of all directed paths from the origin to any q ∈ R d . In orderto do so, we must first consider all “words” in the alphabet consisting of the “directions” given by W .Precisely, let D ( n, N ) be the set of words of length n , in N symbols, with no occurrence of two equalconsecutive symbols appearing in any word. Such words are also known as Smirnov words . In thiscontext, we use the Smirnov words to keep track of the indices of the admissible directions in a givenpath, which each pattern c represents.The moduli space of all directed paths from the origin to q ∈ R d is defined by the union of thefollowing polytopes: M W ( q ) = ∞ a n =0 a c ∈ D ( n,N ) P ( q , c ) This moduli space can be endowed with a natural flat metric, which is one of the innovations in thework of Cano and Díaz. This definition of the muduli space of directed paths also suggests a naturaldefinition for the volume of the moduli space, namely: vol( M W ( q )) := ∞ X n =0 X c ∈ D ( n,N ) vol P ( q , c ) . These definitions also appeared in [2]. To be concrete, we first demonstrate the novel extension ofthe classical binomial coefficients to continuous binomial coefficients. Namely, Cano and Díaz definedfor each q := ( s, x − s ) ∈ R ≥ , with < s < x , the following continuous binomial coefficient:(1.4) (cid:26) xs (cid:27) := vol( M W ( q )) := ∞ X n =0 X c ∈ D ( n, vol P ( q , c ) . We note that here the dimension is d = 2 , the set of admissible directions is W := { (1 , , (0 , } , andso N = 2 . In addition, each path of length n has a pattern c := ( c , . . . , c n ) , and hence the dimensionof the corresponding path polytope P ( q , c ) has dimension n − .Moreover Cano and Díaz obtained the following interesting formula for the continuous binomialcoefficients:(1.5) (cid:26) xs (cid:27) = 2 I (cid:16) p s ( x − s ) (cid:17) + x p s ( x − s ) I (cid:16) p s ( x − s ) (cid:17) , where I ν ( z ) denotes the modified Bessel function of the first kind.We study the d -dimensional extension of this continuous binomial coefficient to continuous multi-nomial coefficients. Suppose we consider all lattice paths from the origin to any q ∈ Z d , using thestandard basis as the set of admissible directions E := { e , . . . , e d } . The classical fact here is that thenumber of such lattice paths equals the multinomial coefficient: (cid:18) q + · · · + q d q , . . . , q d (cid:19) := ( q + · · · + q d )! q ! . . . q d ! . CONTINUOUS ANALOGUE OF LATTICE PATH ENUMERATION 3
We fix any q ∈ R d ≥ , and as before we consider all directed paths between the origin and q . Fixinga pattern c := ( c , . . . , c n ) , we get a path polytope P ( q , c ) with dimension n − d . A natural definitionfor the continuous multinomial would then be:(1.6) (cid:26) x + · · · + x d x , . . . , x d (cid:27) := vol ( M E ( x )) = ∞ X n =0 X c ∈ D ( n,d ) vol P ( q , c ) . However, here we must correct an irregularity in the work of Cano and Díaz. There was a minorerror in their calculation of P ( q , c ) amounting to an extra multiplicative factor of √ d , where d isthe dimension of the ambient space. This leads us to consider a new definition for the continuousmultinomial coefficient, with more details appearing below in Section 2. Assuming this new definitionof a continuous multinomial, we can now state our main results. We recall the Borel transform B ( f ) ,which acts on a univariate function f ( x ) = P ∞ i =0 k i x i (which must therefore be analytic at the origin)by the formula: B ( f )( x ) := ∞ X i =0 k i i ! x i . For a multi-variable analytic function f ( x ) = P ∞ i ,...,i d =0 k i ...i d x i . . . x i d , we define similarly itsBorel transform as B ( f )( x , . . . , x d ) := ∞ X i ,...,i d =0 k i ...i d i ! . . . i d ! x i . . . x i d . Theorem 1.
Let (1.7) F ( x , . . . , x d ) := 11 − (cid:16) x x + · · · + x d x d (cid:17) , which is analytic at (0 , . . . , . Then the continuous multinomial is equal to (cid:26) x + · · · + x d x , . . . , x d (cid:27) = ∂∂x · · · ∂∂x d B ( F )( x , . . . , x d ) . Another interesting result of Cano and Diaz [2] is the following elegant and surprising identity forthe continuous binomial coefficients (in dimension ):(1.8) ∂∂x ∂∂y (cid:26) x + yx (cid:27) = (cid:26) x + yx (cid:27) . This appears the continuous analogue of the usual identity ∆ n ∆ k (cid:18) n + kk (cid:19) = (cid:18) n + kk (cid:19) for binomial coefficients, where ∆ n f ( n ) = f ( n + 1) − f ( n ) is the forward difference operator.We obtain the following generalization of (1.8), in the case of dimension d , for the multinomial coeffi-cients. Theorem 2.
As a multi-variable function, the continuous multinomial satisfies the following partialdifferential equation: (1.9) n Y j =1 (cid:18) ∂∂x j (cid:19) (cid:26) x + · · · + x d x , . . . , x d (cid:27) = n X i =1 Y j = i (cid:18) ∂∂x j (cid:19) (cid:26) x + · · · + x d x , . . . , x d (cid:27) . The paper is organized as follows: in Section 2 we correct an error in [1] and [2], and then motivateour definition of the continuous multinomial. In Section 3 we prove Theorem 1 and derive a closedform expression for the continuous multinomial in terms of the Borel transform. In Section 4 we prove
CONTINUOUS ANALOGUE OF LATTICE PATH ENUMERATION 4
Theorem 2 and extend the two dimensional PDE identity of Cano and Díaz. In Section 5 we showhow to recover discrete multinomials from continuous multinomials, and carry out the calculation intwo dimensions. 2.
A correction and new definitions
Here we point out that the papers of Cano and Diaz [[1], [2]] use a different definition for volumes ofsimplices than the usual Riemannian definition of volume. We now explain this discrepancy betweenthe two definitions of the volume of a simplex. Consider the n -simplex(2.1) ∆ tn := { s , . . . , s n ∈ R ≥ : s + · · · + s n = t } , which is embedded in R n +1 . Now let P tn be the convex hull of the origin and ∆ tn . We note that ∆ n isthe convex hull of the standard basis { e , . . . , e n } while P n is the convex hull of { , e , . . . , e n } . The ( n + 1) dimensional volume of P n is n +1)! , but is also equal to n +1 vol(∆ n ) d (0 , ∆ n ) , where vol is the n -dimensional volume of ∆ n and d (0 , ∆ n ) is the distance from the origin to ∆ n . It can be observedthat d (0 , ∆ n ) = √ n +1 . Therefore,(2.2) vol(∆ n ) = 1( n + 1)! n + 1 d (0 , ∆ n ) = √ n + 1 n ! and(2.3) vol(∆ tn ) = t n √ n + 1 n ! . For instance, when n = 1 , ∆ n is the segment connecting (1 , and (0 , which has length √ = √ .The work [1] instead calculates the volume differently. It uses the following parametrization of ∆ n :(2.4) l = s , l = s + s , . . . , l n = n − X i =0 s i , and writes(2.5) ∆ tn = { ( l , . . . , l n ) ∈ R n : 0 ≤ l ≤ · · · ≤ l n ≤ l n +1 = t } . The paper then claims that vol(∆ tn ) = t n n ! . Although the coordinate change is linear (and is given byan upper triangular matrix) and does not alter the ( n + 1) -dimensional volume on R n +1 , it does changethe induced n -dimensional volume on most affine hyperplanes. The example above demonstrates thispoint for n = 1 .Let T denote the above coordinate change. We note that using T will destroy the product structurewhich is crucial in the computation of the volume of the moduli space of directed paths. More specifi-cally, given a product of two simplices ∆ × ∆ , its image T (∆ × ∆ ) is not a product of two simplices.For example, take ∆ and ∆ to be two intervals ( -simplices). Then their product is a rectangle,which is transformed by T into an parallelogram, which in general is not a product of -simplices.We now discuss what this means for the continuous multinomial case. As a preliminary, the fre-quency vector ν ( c ) of a Smirnov word c encodes the number of times each letter appears in c . Notethat the coordinates of ν ( c ) sum up to n . We now consider(2.6) (cid:26) x + · · · + x d x , . . . , x d (cid:27) := vol ( M E ( x )) := ∞ X n =0 X c ∈ D ( n,d ) vol P ( q , c ) , the previously motivated definition of the continuous multinomial. We take k = d and W = E := { e , . . . , e d } the standard basis of R d . Let ν = ( ν , . . . , ν d ) ∈ Z d ≥ be an integer vector with ν + · · · + CONTINUOUS ANALOGUE OF LATTICE PATH ENUMERATION 5 ν d = n and c a Smirnov word with frequency vector ν . Then, we have the following identities: M c E ( x ) = { a , . . . , a n ∈ R ≥ : a e c + · · · + a n e c n = x } (2.7) = { a , . . . , a n ∈ R ≥ : X i : c i =1 a i = x , . . . , X i d : c id = d a i d = x d } . (2.8)In other words, the vector identity in (2.7) breaks into d independent scalar identities in (2.8). There-fore, M c E ( x ) is the direct product of d simplices isomorphic to ∆ n ( x ) , . . . , ∆ n d ( x d ) , where ∆ m ( y ) := { b , . . . , b m ∈ R ≥ : b + · · · + b m = y } . Stricly speaking, the volume of the simplex ∆ m ( y ) is y m − √ m ( m − .2.1. An alternate definition of volume.
If, instead of the usual Riemannian volume, we would usethe Cano and Diaz modified volume measure, which we call vol CD , and defined by:(2.9) vol CD (∆ m ( y )) = y m − ( m − , then our continuous binomial coefficient would coincide with that of Cano and Díaz. Therefore, bythe product rule of volumes of products of simplices, we would obtain the modified result vol CD ( M c E ( x )) = x ν − ( ν − . . . x ν d − d ( ν d − . and one could interpret this as a definition for the continuous multinomial coefficient, as follows: Definition 3.
Let ν denote the frequency vector of the Smirnov word c . Then(2.10) (cid:26) x + · · · + x d x , . . . , x d (cid:27) := ∞ X n =0 X c ∈ D ( n,d ) x ν − ( ν − . . . x ν d − d ( ν d − . We emphasize that this is the object considered in the Cano and Díaz papers [1] and [2], and in thetwo dimensional case this coincides with the continuous binomial of Cano and Díaz: (cid:26) xs (cid:27) = 2 I (cid:16) p s ( x − s ) (cid:17) + x p s ( x − s ) I (cid:16) p s ( x − s ) (cid:17) . While the continuous multinomial loses some of its geometric intuition and motivation, this renor-malized volume (2.9) leads to an object with very interesting arithmetic properties. In addition, thenon-normalized version with the square root correction terms does not have a closed form in terms ofhypergeometric functions, even in two dimensions.There are two competing definitions for the continuous multinomial, which can lead to some con-fusion. The object (cid:26) x + · · · + x d x , . . . , x d (cid:27) always refers to the definition without a √ n term, so Section 3and 4 refer to the continuous multinomial without the √ n term. However, Section 5 deal directly tothe simplices defining the continuous multinomial. This means that we are using the normal Lebesguemeasure on R n , so we do not use the notation (cid:26) x + · · · + x d x , . . . , x d (cid:27) anywhere. Therefore, if we were tocalculate volumes we would include the √ n term.3. Continuous multinomials
Proof of Theorem 1.
Given a vector ν = ( ν , . . . , ν k ) ∈ Z k ≥ such that ν + · · · + ν N = n , let D ( n, N ; ν ) denote the subset of Smirnov words in D ( n, N ) whose frequency vectors are all equal to ν . As shownby Flajolet and Sedgewick [3], the cardinality of D ( n, N ; ν ) is the coefficient of y ν . . . y ν k N in the powerseries representation of the rational function F ( y , . . . , y N ) = 11 − (cid:16) y y + · · · + y d y N (cid:17) . CONTINUOUS ANALOGUE OF LATTICE PATH ENUMERATION 6
We expand F into a Taylor series about the origin: F ( x , . . . , x d ) = 11 − (cid:16) x x + · · · + x d x d (cid:17) = ∞ X ν ,...,ν d =0 f ν ,...,ν d x ν . . . x ν d d , where f ν ,...,ν d counts the number of Smirnov words with frequency vector ( ν , . . . , ν d ) , as mentionedabove. Therefore, (cid:26) x + · · · + x d x , . . . , x d (cid:27) = ∞ X n =0 X c ∈ D ( n,d ) x ν − ( ν − . . . x ν d − d ( ν d − ∞ X ν ,...,ν d =0 f ν ,...,ν d x ν − ( ν − . . . x ν d − d ( ν d − ∂∂x · · · ∂∂x d ∞ X ν ,...,ν d =0 f ν ,...,ν d x ν ν ! . . . x ν d d ν d ! ! = ∂∂x · · · ∂∂x d B ( F )( x , . . . , x d ) , which completes the proof. (cid:3) When d = 2 , we can set ( x , x ) = ( x, y ) and compute F ( x, y ) = (1 + x + y ) + ∞ X n =1 (cid:0) x n y n + x n y n +1 + x n +1 y n (cid:1) , and B ( F )( x, y ) = (1 + x + y ) + ∞ X n =1 (cid:18) x n n ! y n n ! + x n n ! y n +1 ( n + 1)! + x n +1 ( n + 1)! y n n ! (cid:19) , Therefore, we retrieve the formula for the continuous binomials in [2]: (cid:26) x + yx (cid:27) = (cid:26) x + yx , y (cid:27) = ∂∂x ∂∂y B ( F )( x, y )= ∞ X n =0 (cid:18) x n n ! y n n ! + x n n ! y n +1 ( n + 1)! + x n +1 ( n + 1)! y n n ! (cid:19) = 2 I (2 √ xy ) + ( x + y ) I (cid:0) √ xy (cid:1) √ xy . Partial differential identity
Proof of Theorem 2.
Our approach is inspired by the method of dynamic programming in computerscience.Let us write M ( x ) = M E ( x ) . The moduli space M ( x ) can be decomposed into subspaces M x n +1 ( x ) , . . . , M x d n +1 ( x ) of directed paths of length n + 1 whose last step are in directions e , . . . , e d ,respectively.Consider the subspace M n +1 ( x ) of directed paths of length n +1 from to x following the directionsof the standard basis vectors e , . . . , e d . As above, this subspace can be further decomposed into d pieces M x n +1 ( x ) , . . . , M x d n +1 ( x ) , where M x i n +1 ( x ) is the set of such directed paths whose last step followsthe direction of e i . CONTINUOUS ANALOGUE OF LATTICE PATH ENUMERATION 7
Let ≤ i ≤ d . Suppose the last step is of distance x i − s in direction e i . Then the corresponding sliceof M x i n +1 ( x ) can be identified isometrically with the disjoint union of M x j n ( x , . . . , x i − , s, x i +1 , . . . , x d ) for j = i . Therefore, by Fubini’s theorem, we have vol( M x i n +1 ( x )) = Z x i X j = i vol( M x j n ( x , . . . , x i − , s, x i +1 , . . . , x d )) ds, which implies,(4.1) ∂∂x i vol( M x i n +1 ( x )) = X j = i vol( M x j n ( x )) . The base case is n = d with vol( M x i d ( x )) = ( d − . By convention, for n < d , vol( M x i n +1 ( x )) = 0 .Summing equation (4.1) over all n ≥ d , we obtain the following identity:(4.2) ∂∂x i vol( M x i ( x )) = X j = i vol( M x j ( x )) . or equivalently,(4.3) (cid:18) ∂∂x i (cid:19) vol( M x i ( x )) = d X j =1 vol( M x j ( x )) = vol( M ( x )) . This infers(4.4) n Y j =1 (cid:18) ∂∂x j (cid:19) vol( M x i ( x )) = Y j = i (cid:18) ∂∂x j (cid:19) vol( M ( x )) . Summing this identity over ≤ i ≤ n , we obtain the desired identity. (cid:3) In the case that the dimension d = 2 and ( x, y ) = ( x , x ) , identity (1.9) becomes (cid:18) ∂∂x (cid:19) (cid:18) ∂∂y (cid:19) (cid:26) x + yx (cid:27) = (cid:18) ∂∂x + 1 + ∂∂y (cid:19) (cid:26) x + yx (cid:27) , which simplifies to the Cano and Díaz result ∂∂x ∂∂y (cid:26) x + yx (cid:27) = (cid:26) x + yx (cid:27) . Recovering discrete objects
In this section, we retrieve discrete binomial coefficients from the continuous binomial case. This isdue to a general result in lattice point counting, the
Khovanskii-Pukhlikov theorem . We describe thetheorem, and then carry out a calculation involving it.We begin with the fundamental
Todd operator , which is defined to be the following differentialoperator:
Todd h := d/dh − e − d/dh = X k ≥ ( − k B k k ! (cid:18) ddh (cid:19) k = 1 + 12 ddh + 112 (cid:18) ddh (cid:19) − (cid:18) ddh (cid:19) + . . . Here, B k = B k (0) are the Bernoulli numbers and B k ( x ) are the Bernoulli polynomials with generatingfunction X k ≥ z k B k ( x ) k ! = ze zx e z − . CONTINUOUS ANALOGUE OF LATTICE PATH ENUMERATION 8
We next consider unimodular integral polytopes , i.e. polytopes whose vertices have integercoordinates and whose vertex tangent cones are generated by some basis of Z d (and hence simple).Given a polytope P and vertex v , we define the vertex tangent cone at v to be { v + λ ( y − v ) : y ∈ P, λ ∈ R ≥ } . Suppose P has the hyperplane description P = { x ∈ R d : Ax ≤ b } , where the column vectors of A are primitive integer vectors in Z d . We define the perturbed polytope P ( h ) = { x ∈ R d : Ax ≤ b + h } , for some small h = ( h , h , . . . , h m ) . We also define the multi-dimensional Todd operator Todd h := m Y k =1 Todd h k . The fundamental role of Todd operators is highlighted by the
Khovanskii-Pukhlikov theorem fora unimodular polytope P : P ∩ Z d ) = Todd h vol( P ( h )) | h =0 . More generally, X x ∈ P ∩ Z d exp( x · z ) = Todd h Z P ( h ) exp( x · z ) d x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h =0 . The assumption of unimodularity is important. Loosening it will require replacing the Todd operatorwith much more complicated differential operators, as shown in a version of Euler-Maclaurin formulafor simple polytopes in [5].Lattice paths are directed paths whose steps are only allowed to be integer multiples of one of theadmissible directions. We now consider the binomial case. Given a combinatorial pattern, the modulispace of directed paths with the given pattern, from (0 , to ( x, y ) , is a (direct) product of simplicesof the form ∆ := ∆ xn := { a , . . . , a n ≥ , a + · · · + a n = x } . The space of lattice paths, with the same pattern, from (0 , to ( x, y ) is a (direct) product of discrete simplices of the form Λ := Λ xn := ∆ xn ∩ Z n + = { a , . . . , a n ∈ Z , a , . . . , a n > , a + · · · + a n = x } . Note that the simplex ∆ is not full-dimensional, so the Khovanskii-Pukhlikov theorem does not applydirectly and we do need a small trick to make it work. For a small h = ( h , . . . , h n , h + , h − ) , considerthe perturbed simplex ∆( h ) := { a ≥ h , . . . , a n ≥ h n , x − h − ≤ a + · · · + a n ≤ x + h + } , whose volume is vol(∆( h )) = 1 n ! (( x + h + − h − · · · − h n ) n − ( x − h − − h − · · · − h n ) n ) . We expect
Todd h vol(∆( h )) | h =0 = (cid:18) x − n − (cid:19) . Set h ′ = ( h , . . . , h n ) and use the following notation. ∆ ′ := { a , . . . , a n ≥ , a + · · · + a n ≤ x } , ∆ ′ ( h ′ , h + ) := { a , . . . , a n ≥ , a + · · · + a n ≤ x + h + } , ∆ ′ ( h ′ , h − ) := { a , . . . , a n ≥ , a + · · · + a n ≤ x − h − } , Λ ′ + := { a , . . . , a n ∈ Z , a , . . . , a n > , a + · · · + a n ≤ x } , Λ ′− := { a , . . . , a n ∈ Z , a , . . . , a n > , a + · · · + a n < x } . CONTINUOUS ANALOGUE OF LATTICE PATH ENUMERATION 9
By a polarized version of the Khovanskii-Pukhlikov theorem in [5], we have
Todd ( h ′ ,h + ) vol(∆ ′ ( h ′ , h + )) (cid:12)(cid:12) h ′ =0= h + = ′ + = (cid:18) xn (cid:19) , Todd ( h ′ ,h − ) vol(∆ ′ ( h ′ , h − )) (cid:12)(cid:12) h ′ =0= h − = ′− = (cid:18) x − n (cid:19) . These identities can also be verified manually from the definition of the Todd opearator. Also, it isimportant to note that our simplex is unimodular. Otherwise, the Khovanskii-Pukhlikov theorem doesnot apply, and we will have to use a more complicated version by Karshon-Sternberg-Weitsman [5],which applies to all simple polytopes.Therefore,
Todd h vol(∆( h )) | h =0 = Todd ( h ′ ,h + ) vol(∆ ′ ( h ′ , h + )) (cid:12)(cid:12) h ′ =0= h + − Todd ( h ′ ,h − ) vol(∆ ′ ( h ′ , h − )) (cid:12)(cid:12) h ′ =0= h − = ′ + − ′− = (cid:18) xn (cid:19) − (cid:18) x − n (cid:19) = (cid:18) x − n − (cid:19) = . Further remarks
In general, applying the Khovanskii-Pukhlikov theorem to the simplices that define our continuousobjects will recover the appropriate discrete objects. The Khovanskii-Pukhlikov machinery also worksin the case that the polytope is a simple polytope [5], not merely a unimodular polytope. Thus onecould apply our approach to simple polytopes in order to discretize them and perhaps obtain futurediscretization results of this flavor.In principle, one could begin with an arbitrary set of admissible directions which are not evennecessarily integer vectors, and develop an analogous theory.Another interesting direction for future research is the pursuit of an L -metric approach to volumes,since incorporating such an L approach into the Cano-Diaz machine might yield very interestingresults. References [1] L. Cano and R. Díaz, Indirect Influences on Directed Manifolds. ArXiV:1507.01017v4[math-ph].[2] L. Cano and R. Díaz, Continuous analogues for the binomial coefficients and the Catalan numbers.ArXiV:1602.09132v4[math.CO].[3] P. Flajolet, R. Sedgewick, Analytic Combinatorics. Cambridge University Press, 2009.[4] I. S. Gradshteyn and I. M. Ryzhik, eds., Table of Integrals, Series, and Products. 7th ed., Academic Press, San Diego,2007.[5] Y. Karshon, S. Sternberg and J. Weitsman, Exact Euler–Maclaurin formulas for simple lattice polytopes. Advancesin Applied Mathematics, Volume 39, 2007, 1–50.[6] A. D. Kolesnik, Moment analysis of the telegraph random process, Buletinul Academiei De Stiinţe a RepubliciiMoldova. Matematica, Number 1(68), 2012, 90–107.[7] A. P. Prudnikov, Y. A. Brychkov and O.I. Marichev, Integrals and Series, Volume 2, Gordon and Breach SciencePublishers, 1986.[8] W.T. Ross and H.S. Shapiro, Generalized Analytic Continuation, A.M.S., University Lecture Series 25, 2002. † University of Maryland, College Park, MD 20742, USA † National Institute for Biological and Mathematical Synthesis, Knoxville, TN 37996, USA
E-mail address : [email protected] ‡ Tulane University, New Orleans, LA 70118, USA
CONTINUOUS ANALOGUE OF LATTICE PATH ENUMERATION 10 ‡ LSS/Supelec, Universite Paris Sud Orsay, France
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