Higher Dimensional Lattice Chains and Delannoy Numbers

Transcription

1 Linfield College Faculty Publications Faculty Scholarship & Creative Works -- Higher Dimensional Lattice Chains and Delannoy Numbers John S Caughman Portland State University Charles L Dunn Linfield College Nancy Ann Neudauer Pacific University Colin L Starr Willamette University Follow this and additional works at: Part of the Discrete Mathematics and Combinatorics Commons DigitalCommons@Linfield Citation Caughman, John S; Dunn, Charles L; Neudauer, Nancy Ann; and Starr, Colin L, "Higher Dimensional Lattice Chains and Delannoy Numbers" ( Faculty Publications Accepted Version Submission This Accepted Version is brought to you for free via open access, courtesy of DigitalCommons@Linfield For more information, please contact digitalcommons@linfieldedu

2 HIGHER DIMENSIONAL LATTICE CHAINS AND DELANNOY NUMBERS JOHN S CAUGHMAN, CHARLES L DUNN, NANCY ANN NEUDAUER, AND COLIN L STARR Abstract Fix nonnegative integers n,,, and let L denote the lattice of points (a,, a d Z d that satisfy a i n i for i d Let L be partially ordered by the usual dominance ordering In this paper we use elementary combinatorial arguments to derive new expressions for the number of chains and the number of Delannoy paths in L Setting n i = n (for all i in these expressions yields a new proof of a recent result of Duichi and Sulanke [9] relating the total number of chains to the central Delannoy numbers We also give a novel derivation of the generating functions for these numbers in arbitrary dimension Introduction Lattice chains and Delannoy paths have commanded a great deal of attention historically, and have enjoyed a surge of interest in recent decades Popular expositions of the subject, like Comtet [8] and Stanley [5], have certainly given further impetus to their study, while also providing powerful tools for their analysis At the same time, interest has also been generated with the appearance of connections to other topics, as chains have been studied in relation to simplicial complexes, Legendre polynomials, formal languages, ballot numbers, and probability theory, to name only a few [, 6, 4, 4, ] The interested reader can find even more on these topics in the survey by Banderier and Schwer [], with over 75 bibliographic references A particular charm of the topic is the interplay between counting arguments and generating function techniques In Stanley [5], a problem involving lattice chains and Delannoy paths in two dimensions was used to illustrate a technique for extracting the diagonal of a generating function Specifically, in the special case when the -dimensional lattice is square, the number of chains exceeds the number of Delannoy paths by a factor of an appropriate power of The question was then posed to find a combinatorial proof of the same result This challenge was met by Sulanke in [6], who established a bijective correspondence by composing a sequence of intermediate bijections between six different step sets in the -dimensional lattice for the central (diagonal case More recently, his article with Duchi [9] generalizes the result to the central case in arbitrary dimension, again by means of a composition of explicit bijections In the present paper, we offer elementary counting techniques that yield a number of new expressions, both for chains and Delannoy paths in the general (not necessarily central lattice, in any dimension The expressions for the chains and for the Delannoy numbers are strikingly similar to each other, and upon an appropriate subtitution, the central Date: September 8,

3 JOHN S CAUGHMAN, CHARLES L DUNN, NANCY ANN NEUDAUER, AND COLIN L STARR lattice is obtained as a special case, yielding an alternate proof of Sulanke s theorem in any dimension The chain numbers and the Delannoy numbers satisfy similar cross-dimensional recurrence relations, and we exploit this recursive structure to prove a result which generalizes both recurrences and offers a new means to derive their generating functions easily and uniformly in any dimension The paper is organized as follows Section fixes notation and describes our results on lattice chains, including k-chains, reducible chains, and finally the total number of chains In Section, we consider Delannoy paths, first with k steps, and then the general case, obtaining the desired expression related to the number of chains Finally, in Section 4, we introduce the class of a-recurrent sequences of functions a class that includes both the chain numbers and the Delannoy numbers as special cases and we offer an explicit expression for their generating functions in any dimension Results on Lattice chains Throughout this paper, N denotes the nonnegative integers and P the positive integers Fix d P and n N d, where n = (n,, T Let L(n denote the lattice of integer points (a,, a d T N d satisfying a i n i for i d Recall L(n is partially ordered by the dominance relation, defined as follows Given a, b L(n with a = (a,, a d T and b = (b,, b d T, we say a b whenever a i b i for each i ( i d We write a b whenever a b and a b Define the weight of an element a = (a,, a d T L(n by wt(a = a + +a d We define the truncation of a to be the (d -tuple a = (a,, a d T Counting k-chains and some variations By a chain in L(n we mean a subset of L(n that is totally ordered by A k-chain is a chain with k elements Let C(n denote the set of chains in L(n, and for each integer k, let C k (n denote the set of k-chains in L(n In this section we study expressions for C k (n and C(n Expressions for C k (n are not difficult to derive, and have been computed in several places [, ] for the special case n i = for all i, and, in [7], for the general case Each of these derivations proceeds either by solving an appropriate recurrence or through the use of generating functions In [5], a direct counting argument was given for C k (n using the principle of inclusion/exclusion Lemma [5],[7] Fix n N d, where n = (n,, T and for each k N, let C k (n denote the set of k-chains in the corresponding lattice L(n, and C k (n the set of chains in C k (n that contain the maximum element n Then the following hold: (i The maximum length of a chain in L(n is given by k max = wt(n + (ii For any integer k ( k k max, the number of k-chains in L(n is given by k ( d ( k C k (n = ( r ni + k r r r= i= n i

4 HIGHER DIMENSIONAL LATTICE CHAINS AND DELANNOY NUMBERS (iii For any integer k ( k k max, the number of k-chains in L(n that contain n is given by C k (n = k ( i+ C k i (n i= Proof (i If a and b are elements of L(n such that a b, then wt(a < wt(b wt(n Since the weight of any element must be an integer, a chain can have at most k max = wt(n + elements Conversely, a chain with length k max can easily be defined inductively as follows We simply set a =, and, given any a i with i k max, we define a i+ by adding to any coordinate a ij of a i for which a ij < (ii An elementary proof using inclusion/exclusion is given in [5] (iii Note that C (n = and C (n = For k, each k-chain containing n corresponds to a unique (k -chain that does not contain n (and conversely So C k (n = C k (n \ C k (n = C k (n C k (n The result now follows by a simple induction Counting reducible chains We say a chain ξ is reducible if the truncations of its elements are pairwise distinct Recall that chains are not defined as sequences, but as subsets of the lattice, so a chain cannot contain repeated elements With this in mind, we could equivalently define a k-chain ξ in L(n to be reducible iff the set ξ, formed by truncating the elements of ξ, remains a k-chain in L(n For example, let n = (, 4,, 4 T and suppose ξ and ξ denote the -chains ξ : and ξ : Then ξ is not reducible, since the first two elements have identical truncations Equivalently, we could say that ξ is not reducible since ξ is only a -chain in L(n, as shown below On the other hand, ξ is reducible, since ξ is still a - chain ξ : and ξ : The next result is the analog of Lemma for reducible chains Lemma With the notation of Lemma, let C red (n denote the set of reducible chains in L(n, and let C red (n denote the set of reducible chains that contain n Then the following hold (i The maximum length of a chain in C red (n is k max := wt(n + (ii The number of reducible chains in L(n is C red (n = k max k= ( nd + k C k (n

5 4JOHN S CAUGHMAN, CHARLES L DUNN, NANCY ANN NEUDAUER, AND COLIN L STARR (iii The number of reducible chains in L(n that contain n is C red (n = k max k= ( nd + k C k (n Proof (i By truncating the d-coordinates of each element, every reducible k- chain ξ in L(n corresponds to a unique k-chain ξ in L(n Therefore, k k max by Lemma (i (ii Fix an integer k ( k k max and let ξ be any reducible k-chain Truncating the d-coordinates of the elements of ξ gives a unique k-chain ξ in L(n, and the d-coordinates themselves form a non-decreasing sequence σ of integers between and (inclusive Conversely, such a sequence and a k-chain in L(n correspond to a unique reducible chain in L(n The number of such sequences is ( +k Multiplying by C k (n and summing over k, we obtain the result red (iii As in (ii above, each ξ in C k (n corresponds to a unique ξ in C k (n and a non-decreasing sequence σ of integers between and (inclusive, where σ contains at least once The number of such sequences is ( +k Multiplying by C k (n and summing over k, we obtain the result Corollary With the notation of Lemma, the number of reducible chains in L(n that contain n is given by C red (n = k max k k= i= ( i+ ( nd + k C k i (n Proof Immediate by Lemma (iii and Lemma (iii By Lemma (ii, we can evaluate the term C k i (n in the expression in Corollary above to obtain a triple sum As the next corollary shows, however, this reduces to a double sum Corollary With the notation of Lemma, the number of reducible chains in L(n that contain n is given by C red (n = k max k k= i= ( i+k ( k i ( nd + k d ( nj + i Proof Consider the expression for C red (n given in Corollary above Recall that C (n =, and for i < k we can evaluate C k i (n using Lemma (ii to obtain j= ( C red (n = k max k= ( [ nd + k ( k+ k + i= k i r= ( r+i+ ( k i r d ( ] nj + k i r j=

6 HIGHER DIMENSIONAL LATTICE CHAINS AND DELANNOY NUMBERS 5 With the change of variables r = k i t, this simplifies to ( C red (n = k max ( k nd + k k i ( k i ( k+ + ( t t k= i= t= d Interchanging the order of summation over i and t, this is equivalent to ( C red (n = k max ( k nd + k ( k+ + k= t= ( t d ( nj + t j= k t ( nj + t j= ( k i t i= ( k i i= t A common binomial identity [, Thm 8] states that k t ( = k t Applying this identity and then substituting t = i, the bracketed expression simplifies to give the desired result The total number of chains Keeping with the notation of Lemma, we let C(n denote the set of chains in L(n that contain n It is convenient to count C(n rather than C(n directly The difference is minimal, however, since removing n from each chain in C(n gives a bijection between C(n and C(n \ C(n, so that (4 C(n = C(n Let P denote the power set of {,,, }, and recall that C red (n denotes the set of reducible chains in L(n that contain n In this section we establish a bijection φ between C(n and P C red (n Roughly speaking, φ can be described as follows Given a chain ξ that contains n, it fails to be reducible if the truncations of its elements are not distinct The function φ removes from ξ any elements whose truncations are repeated by a later element in ξ Doing so produces a reducible chain ξ red The d-coordinates of the elements removed are recorded in a set A ξ The output of φ is the pair (A ξ, ξ red More formally, we have the following Definition Suppose a chain ξ in C(n has k elements a a k, where a i = (a i,, a id T for each i ( i k We define A ξ = {a id a i = a i+}, and ξ red = ξ \ {a i a i = a i+}, and we let φ(ξ denote the pair (A ξ, ξ red To illustrate this definition, let n = (,, T and suppose ξ denotes the following 8-chain in C(n: (5 ξ : a a a a 4 a 5 a 6 a 7 a 8 ( ( Notice that a = a = a 4 = and a 6 = a 7 = The reducible chain ξ red is formed by removing a and a (keeping a 4, and removing a 6 (keeping a 7

7 6JOHN S CAUGHMAN, CHARLES L DUNN, NANCY ANN NEUDAUER, AND COLIN L STARR For each of the elements removed, their last coordinates ( rd coordinates in this case are recorded in the set A ξ Then ξ red is a reducible 5-chain in C red (n, the set A ξ is a subset of {,, }, and φ(ξ denotes the pair (A ξ, ξ red below: (6 A ξ = {,, } and ξ red : Observe that φ(ξ P C red (n Next we describe how the original chain ξ can be recovered from the pair (A ξ, ξ red Given the information in line (6 above, we simply must reinsert into ξ red the missing elements, one belonging to each member of A ξ Each x in A ξ is the d-coordinate x d of a point x that is to be inserted immediately to the leftof the first y in ξ red for which x d < y d In our case, and belong left of, while belongs left of : Observe that for each x in A ξ, such a y is guaranteed to exist in ξ red by the fact that every element of A ξ is <, while n belongs to ξ red Indeed, this motivates our choice to work with C(n rather than C(n To complete the recovery of ξ, note that the remainder of each new point x is determined by the condition that ( x = y In our case, and are topped by Doing so produces the original chain ξ, given in (5 The next three lemmas establish the relevant properties of φ, while is topped by Lemma With the above notation, φ is a function from C(n to P C red (n Proof For ξ C(n, recall φ(ξ = (A ξ, ξ red If x A ξ, then x = a id for some i, where a i = a i+ But a i a i+, so a id < a i+,d Thus every element of A ξ is strictly less than, and A ξ P To show ξ red C red (n, note that ξ red ξ, so ξ red is totally ordered by And n ξ since ξ C(n, while n {a i a i = a i+ } so n ξ red It remains to show ξ red is reducible Suppose there were x y in ξ red such that x = y Then x = a i and y = a j for some i < j But a i a i+ a j so a i = a i+ and thus a i ξ red, a contradiction It follows that ξ red C red (n Lemma 4 With the above notation, φ is injective Proof Let ξ, ξ be in C(n and suppose φ(ξ = φ(ξ Then ξ red = ξ red, and to prove ξ = ξ, it remains to show that ξ \ ξ red = ξ \ ξ red We accomplish this by proving that for any chain ξ in C(n, each element x ξ \ ξ red corresponds to a unique element x d A ξ, and that, in fact, x can be explicitly constructed from the element x d A ξ and the chain ξ red Performing this construction for each element of A ξ then yields the entire set ξ \ ξ red To describe the construction, let x be any (

8 HIGHER DIMENSIONAL LATTICE CHAINS AND DELANNOY NUMBERS 7 element of ξ \ ξ red, and let k denote the length of ξ Then x = a i for some i where a i = a i+ and x d = a id A ξ Let t = max{j a i = a j } Then t i + and a i = a i+ = = a t Also, either t < k and a t a t+ or else t = k and a t = n In either case, a t ξ red, and a i,, a t ξ red So a t = min{y y ξ red and x y} Observe that since x a t and x = a t, it must be the case that x d < a td It follows that a t = min{y y ξ red and x d < y d } Since x = a t and has d-coordinate x d, we have now shown that x is completely determined by the element x d in A ξ and the chain ξ red It follows that ξ is determined by the pair (A ξ, ξ red, so φ is injective Lemma 5 With the above notation, φ is surjective Proof To see that φ is surjective, we associate a chain in C(n with each pair (A, ζ in P C red (n Let (A, ζ be such a pair and suppose ζ has t elements b b t where b i = (b i,, b id T for each i ( i t For each x in A, define m := min{j x < b jd } and set b x = (b m,, b m(d, x T in L(n In other words, we define b x by putting b x := b m and setting the d-coordinate of b x equal to x Then the chain in C(n that we associate with the pair (A, ζ is simply ξ (A,ζ := ζ {b x x A} It is easy to check that φ(ξ (A,ζ = (A, ζ as desired Corollary With the above notation, the map φ is a bijection between C(n and P C red (n Proof Immediate from Lemmas -5 Theorem Fix n N d and let C(n denote the set of chains in L(n Then k max C(n = + k k= i= ( i+k ( k i ( nd + k ( nj + i d Proof By equation (4 and Corollary, we have C(n = P C red (n = P C red (n Since P =, the result follows by Corollary j= Results on Delannoy numbers and the Theorem of Sulanke The set D = D(n of (generalized Delannoy paths contains precisely those chains in L(n that contain both the origin = (,, T and n = (n,, T and whose successive elements differ by at most one in each coordinate In other words, the elements of D(n correspond to walks from to n in which only positive steps from the d-dimensional unit hypercube are allowed This follows [] The cardinalities D(n are referred to as (generalized Delannoy numbers For more about generalizations of the Delannoy numbers, we refer the reader to [, ] When all the n i share a common value n, we have n = (n,, n T and we refer to the cardinalities D(n as the (d-dimensional central Delannoy numbers In this section we use an inclusion/exclusion argument to find an expression for the general Delannoy numbers which specializes to a useful expression for the central Delannoy numbers in Theorem 4

9 8JOHN S CAUGHMAN, CHARLES L DUNN, NANCY ANN NEUDAUER, AND COLIN L STARR Delannoy paths with k steps It is common to refer to the size of a Delannoy path by the number of steps it contains, rather than the number of elements it has as a chain in L(n In other words, suppose a chain ξ in D(n has elements a a a k Then we say ξ has k steps (Notice that ξ has k + elements, and hence, length k + as a chain The set of all k-step Delannoy paths is denoted by D k (n Due to symmetry, the ordering of the dimensions in L is often irrelevant, so we frequently assume that n n Under this assumption, it is easy to show that the minimum number of steps a Delannoy path can have is, while the maximum is n + +, which corresponds to k max from earlier in this article Finally, we remark that, since each k-step Delannoy path begins and ends with the points a = and a k = n, we could equivalently represent a path a a a k by the sequence b, b,, b k, where each b i = a i a i In this representation, the b i are nonzero d-tuples of s and s This observation is the key to the following result Theorem Fix n N d such that n n Let k max = n Then, for each k ( k k max, the number of k-step Delannoy paths in the lattice L(n is given by ( k D k (n = k nd i= ( i ( k nd i ( k i d Proof Observe that each k-step Delannoy path a a a k corresponds uniquely to a sequence B = b, b,, b k, where each b i = a i a i Each b i is a nonzero d-tuple (b i, b i,, b id T of s and s By the definition of a Delannoy path, projection of B onto the j-coordinate must give a sequence B j = b j, b j,, b kj of s and s that contains precisely ones, for each j ( j d We count the number of such sequences B as follows First, we choose the sequence B d = b d, b d,, b kd of ones and k zeros There are ( k choices for B d Next, we must choose sequences B j for j d in such a way that each sequence has exactly ones, but we must also ensure that the resulting sequence B has no zero terms This amounts to ensuring that, for each zero term in B d, there is at least one B j which is nonzero in that term In other words, for each i where b id =, there must be at least one j ( j d for which b ij = This is achieved by the method of inclusion/exclusion, as follows Let Z = {i b id = }, and for each T Z, let s(t be the number of sequences S = s, s,, s k, such that all of the conditions (i-(iv hold below (i For each i ( i k, the term s i is a d-tuple (s i, s i,, s id T of s and s; (ii The d-projection of S satisfies S d = B d (iii Each j-projection S j = s j, s j,, s kj has precisely ones (iv For each t T, the term s t = To count s(t, we note that condition (ii fixes the d-projection of S It remains to satisfy (iii by forming d independent sequences S j, each with the specified number of ones, and subject to the constraint (iv that s tj = for t T So for each j, there are ( k t choices for such a sequence, where t = T Taken together, we arrive at the total s(t = d ( k t j= j=

10 HIGHER DIMENSIONAL LATTICE CHAINS AND DELANNOY NUMBERS 9 Now, since T can range from to k, the number of sequences B that have the specified d-projection B d and contain no zero terms is, by inclusion/exclusion, k n d ( d k ( T s(t = ( t nd ( k t t T Z Recalling the number of choices for B d was ( k, we obtain the result t= The total number of Delannoy paths Theorem Fix n N d such that n n and let k max = n Then the total number of Delannoy paths in the lattice L(n is given by D(n = k max k k= i= ( i+k ( k i ( nd + k j= d ( nd + i Proof To find D(n, we sum the expression for D k (n from Theorem over all k from to k max to obtain: k max ( k nd k ( d k D(n = ( i nd ( k i i k= i= Reindexing the outer sum, this simplifies to k max ( k nd + k ( k D(n = ( i i k= i= d j= j= ( nd + k i Replacing i by k i reverses the order of the inner sum, and simplification gives the desired result j= Specializing to the central case, we have the following Theorem 4 [9] Fix n N d such that n i = n for all i ( i d Then C(n = n+ D(n Proof Immediate from Theorems and 4 Generating Functions For dimensio =, the total number of chains C(n, n satisfies the recurrence C(n, n = C(n, n + C(n, n C(n, n for n, n, and C(n, n = n+n whenever n n = Similarly, the total number of Delannoy paths D(n, n satisfies the recurrence D(n, n = D(n, n + D(n, n + D(n, n for n, n, and D(n, n = whenever n n = Using these recurrences, the generating functions can be derived Let g C (x, y and g D (x, y be the generating functions for C(n, n and D(n, n, respectively So g C (x, y = x + y xy,

11 JOHN S CAUGHMAN, CHARLES L DUNN, NANCY ANN NEUDAUER, AND COLIN L STARR and g D (x, y = x y xy Generalizing these results to higher dimensions requires some additional notation For any positive integer d, let [d] denote the set {,,, d} Also, let B d = {, } d \ {} Recall the support of a N d is supp(v = { j [d] a j } Observe that for v B d we have that wt(v = supp(v, which is nonzero by the definition of B d Let n N d with n = (n, n,, If T = { i, i,, i m } is a nonempty subset of [d], then we define n T to be the vector of length d m obtained by removing the i, i,, i m components of n When T = we set n T = n For arbitrary dimensio it is easily verified that for n P d we have C(n = ( wt(v+ C(n v, v B d and D(n = D(n v v B d Because these recurrences are similar in form, we treat them both as special cases of a more general form to derive the generating functions uniformly We begin with the following definition Definition Let a = a, a, a, be a sequence of real numbers A sequence of functions F = F, F, F,, where F d : N d R for all d P, is a-recurrent if the following (i-(iii hold (i For all d P and n P d, F d (n = v B k a wt(v F d (n v (ii For all d P and nonzero n N d \ P d, where T := [d] \ supp(n (iii For all d P, F d (n = F d T (n T, F d ( = a It is not difficult to see that for a given sequence a = a, a, a,, there is a unique sequence of functions which is a-recurrent To illustrate this definition, let F d (n = C(n, where n N d So the sequence of chain numbers is a-recurrent with a =,,,,,,, where the signs alternate after the first two entries Similarly, let F d (n = D(n, where n N d So the sequence of Delannoy numbers is a-recurrent with a =,,, Theorem 5 Let F = F, F, F, be an a-recurrent sequence of functions with a = a, a, a, For d P, let g d (x = g d (x, x,, x d be the generating function for F d Then g d (x = a a S x i =S [d] i S

12 HIGHER DIMENSIONAL LATTICE CHAINS AND DELANNOY NUMBERS Proof We proceed by induction o If d =, then F (n = a F (n for n P by Def (i and F ( = a by Def (iii So F (nx n = a F (n x n n= n= Thus g (x a = xa g (x giving that g (x = a ( a x, as desired Now fix an integer d and suppose that the statement holds for all j [d ] Using the fact that our sequence of functions is a-recurrent, we have that (7 F d (nx n xn x d = a wt(v F d (n vx n n P d n P d v B d Consider the left-hand side of (7 a-recurrent, we have (8 F d (nx n xn x d n P d xn x d By inclusion-exclusion and the definition of = ( d a + ( S g d S (x S S [d] Now consider the right-hand side of (7 Let v B d and let S v := supp(v = { i, i,, i wt(v } Note that wt(v Again by inclusion-exclusion, for this particular v, we have (9 n P d a wt(v F d (n vx n xn x d = a wt(v x i x i x iwt(v T [d]\s v ( T g d T (x T By (7, the expression on the right side of (8 must equal the sum over all v B d of the expression on the right side of (9, giving ( ( d a + ( S g d S (x S = S [d] v B d a wt(v x i x i x iwt(v T [d]\s v ( T g d T (x T Each v B d corresponds to a unique nonempty subset V [d] and conversely, so ( can be rewritten as ( ( d a + ( S g d S (x S = a V ( T g d T (x T S [d] =V [d] i V x i T [d]\v Swapping the order of summation on the right yields ( ( d a + ( S g d S (x S = ( T g d T (x T S [d] T [d] =V [d]\t a V x i i V

13 JOHN S CAUGHMAN, CHARLES L DUNN, NANCY ANN NEUDAUER, AND COLIN L STARR Collecting all instances of g d (x on the left side, we obtain ( g d (x a V x i = ( d+ a =V [d] i V =S [d] + ( S g d S (x S =T [d] ( T g d T (x T =V [d]\t a V x i It remains to show that the right side equals a To this end, observe that the two sums on the right can be combined Doing so reduces the right side of the above to ( d+ a ( S g d S (x S a V =S [d] By the induction hypothesis, g d S (x S = a =V [d]\s =V [d]\s a V Therefore, substitution into ( gives us g d (x (4 a V = ( d+ a (5 =V [d] i V x i i V x i =S [d] i= i V x i ( S a i V d ( ] d = a [( d+ ( i i The bracketed expression equals, by a well-known identity [, Th 7], and the result now follows Corollary 4 Let g D (x, y, z be the generating function for the -dimensional Delannoy numbers Then g D (x, y, z = x y z xy xz yz xyz Corollary 5 Let g C (x, y, z be the generating function for the -dimensional chain numbers Then g C (x, y, z = (x + y + z xy xz yz + xyz References [] I Anderson A First Course in Discrete Mathematics Springer, London, [] J-M Autebert and S R Schwer On generalized Delannoy paths SIAM J Discrete Math, 6(:8, [] C Banderier and S Schwer Why Delannoy numbers? Journal of Statistical Planning and Inference, 5(:4 54, 5 [4] J Bertoin and J Pitman Path transformations connecting Brownian motion, excursion and meander Bulleties Sciences Mathématiques, 8(:47 66, 994 [5] J S Caughman, C R Haithcock, and J J P Veerman A note on lattice chains and Delannoy numbers Discrete Math, 8:6 68, 8

14 HIGHER DIMENSIONAL LATTICE CHAINS AND DELANNOY NUMBERS [6] G-S Cheon and E-M Moawwad Extended symmetric Pascal matrices via hypergeometric functions Appl Math Comput, 58(:59 68, 4 [7] W-S Chou Formulas for counting chains in multisets Utilitas Mathematica, 4:, 99 [8] L Comtet Advanced Combinatorics D Reidel Publishing Co, Dordrecht, 974 [9] E Duchi and R A Sulanke The n factor for multi-dimensional lattice paths with diagonal steps Séminaire Lotharingiee Combinatoire, 5, 4 Article B5c [] H W Gould and M E Mays Counting chains of subsets Utilitas Mathematica, :7, 987 [] G Hetyei Central Delannoy numbers and balanced Cohen-Macaulay complexes Annals of Combinatorics, :44 46, 6 [] S Kaparthi and H R Rao Higher dimensional restricted lattice paths with diagonal steps Discrete Applied Mathematics, :79 89, 99 [] R B Nelson and Jr H Schmidt Chains in power sets Mathematics Magazine, 64:, 99 [4] S R Schwer S-arrangements avec répétitions C R Acad Sci Paris, 4: 6, [5] R P Stanley Enumerative Combinatorics, Volume Cambridge University Press, Cambridge, 999 [6] R A Sulanke Counting lattice paths by narayana polynomials The Electronic Journal of Combinatorics, 7(: 9, Department of Mathematics & Statistics, Portland State University, Box 75, Portland, OR 97 Department of Mathematics, Linfield College, 9 SE Baker Street, McMinnville, OR 978 Department of Mathematics, Pacific University, Forest Grove, OR 976 Department of Mathematics, Willamette University, Salem, OR 97