We obtain formulas for the growth rate of the numbers of certain paths in
infinite graphs built on the two-dimensional Eulerian graph. Corollaries are
identities relating Stirling numbers of the first and second kinds.Comment: Misprint corrected. To appear in Proc. Amer. Math. So

Petersen, K.

Varchenko, A.

English

arXiv

We obtain formulas for the growth rate of the numbers of certain paths in a multi-dimensional analogue of the Eulerian graph. Corollaries are new identities relating Stirling numbers of the first and second kinds

Carolina Digital Repository

arXiv:0911.1970v2  [math.CO]  24 Feb 2011PATH COUNT ASYMPTOTICS AND STIRLING NUMBERSK. PETERSEN AND A. VARCHENKOAbstract. We obtain formulas for the growth rate of the numbers of certainpaths in a multi-dimensional analogue of the Eulerian graph. Corollaries arenew identities relating Stirling numbers of the first and second kinds.1. IntroductionA graded, infinite, directed graph defines a dynamical system called a Bratteli-Vershik or adic system (see [8] and the references given there for background onadic systems). One of the main problems concerning these systems is to determinetheir ergodic invariant measures, and the basic tool for attacking this problem isthe asymptotics of numbers of paths in the graph with certain properties.A very interesting Bratteli-Vershik dynamical system is associated to the Euler-ian two-dimensional graph with vertices at the points of Z2≥0. The paths in theEulerian graph may be counted in many different combinatorial ways, but all at-tempts to extend those counts to the multi-dimensional analogues of the Euleriangraph have failed up to this moment; techniques applied to the two-dimensionalEulerian graph, such as the recurrence of random walks in two dimensions in [1],the coding of paths by permutations in [5], or the left-to-right monotonicity in [7],are not available in dimensions higher than two.In this paper we make a first step in the multi-dimensional direction. We considera multi-dimensional analogue of the Eulerian graph and determine the asymptoticsof the numbers of paths with certain properties. We plan to apply these asymptoticsto studying the ergodic measures on these and related systems in future work. Asa rather unexpected corollary of our asymptotics we get a family of new identitiesbetween Stirling numbers of the first and second kinds, which are classical objects,arise in a variety of combinatorics problems, and for which many general relationshave long been known (see for example [9, 10]).Date: August 29, 2018.1991 Mathematics Subject Classification. Primary: 05A10, 05A16, 05A19, 05C30, 05C63. Sec-ondary: 37A05, 37A50.Key words and phrases. Eulerian numbers, Stirling numbers, symmetric polynomials, rein-forced random walks, urn models.The research of the second author was supported in part by NSF grant DMS-0555327.12 K. PETERSEN AND A. VARCHENKONow we formulate the results of this paper. Denote by e1 = (1, 0, . . . , 0), . . . ,en+1 = (0, . . . , 0, 1) ∈ Zn+1 the ordinary basis vectors in Rn+1. Let c ∈ Zn+1≥0 .We define the graph Ec to be the graph with vertices i ∈ Zn+1≥0 and the followingdirected multiple edges. Each vertex i is connected just to n+1 vertices: to vertexi + ej by cj + in+1 edges if j < n+ 1 and to i + en+1 by cn+1 + i1 + i2 + · · · + inedges.Such a graph models a random walk on the graph Gn+1 consisting of n+1 loopsbased at a single vertex, with a special kind of opposite reinforcement: when thewalker takes a step in the direction of en+1, the numbers of edges in all the otherdirections (ej for j = 1, . . . , n) are incremented by 1; but when the walker takes astep in the direction of ej for some j = 1, . . . , n, only the number of edges in thedirection of en+1 is incremented by 1 (cf. [6]). When n = 1 and c = (1, 1), Ec isthe standard two-dimensional Eulerian graph. The corresponding two-dimensionalEulerian adic system was introduced in [1] and studied further in [5] and [8]; seealso [2, 4, 7].For each i ∈ Zn+1≥0 denote by Ac(i) the number of paths in the graph Ec from0 to i. The standard Eulerian numbers (see [3, 8]) are the numbers A(i1, i2) =A(0,0)(i1, i2) with n = 1. In this paper, we give a formula for the leading term ofthe growth rate of Ac(i1, . . . , in, in+1) as in+1 tends to infinity. We show that ifm = i1 + · · ·+ in, thenAc(i1, . . . , in, in+1) ∼ Bc(i1, . . . , in)(cn+1 +m)in+1as in+1 → ∞ and(1.1) Bc1,...,cn+1(i1, . . . , in) =(cn+1 +m)mm!∑f∈Fm∏j=1(cf(j) + cn+1 + j − 1) ,where F denotes the set of all maps f : {1, . . . ,m} → {1, . . . , n} such that |f−1(j)| =ij for j = 1, . . . , n.As a corollary we obtain new identities relating Stirling numbers of the first andsecond kinds; for 1 ≤ k ≤ n, 0 ≤ r ≤ k, we get(1.2)(r + n− k − 1r)s1(n, r + n− k) =k∑m=0(m+ n− km+ 1) r∑i=0(i + n− k +m− 1i)(−1)m+r−inr−i+1(r − i+ 1)! ×s2(m+ 1, r − i + 1)s1(n, i+ n− k +m),where s1, s2 are Stirling numbers of the first and second kinds: see Theorem 3.10and Corollary 3.11.Acknowledgment. We thank R. Stanley for telling us about theorem of Frobenius(Theorem 3.6) and the referee for suggestions leading to the improvement of theexposition.PATH COUNTS AND STIRLING NUMBERS 32. Path counts and limits2.1. Finding the leading term. The numbers Ac(i) of paths in the graph Ecfrom 0 to i satisfy the recurrence relation(2.1) Ac(i) = (cn+1 + i1 + · · ·+ in)Ac(i− en+1) +n∑j=1(cj + in+1)Ac(i − ej).We are interested in the limit(2.2) Bc(i1, . . . , in) = limin+1→∞Ac(i1, . . . , in, in+1)(cn+1 + n)in+1.Proposition 2.1. Let u = (1, . . . , 1) ∈ Zn≥0. Then(2.3) Bc(u) =∑ξ∈Sn∞∑j1,...,jn=0n∏k=1(cξ(k) + j1 + · · ·+ jk)(cn+1 + k − 1cn+1 + n)jk.Proof. To form a path from 0 to i = (1, . . . , 1, in+1) in Ec, we need to take one stepin the direction of each ek, k = 1, . . . , n, and in+1 steps in the direction of en+1.Thus each such path will correspond to a permutation ξ of 1, 2, . . . , n which willdetermine when the step in the direction of each ek, k = 1, . . . , n, is taken, togetherwith a partition in+1 = j1+ · · ·+ jn+ jn+1 of in+1 into nonnegative integers, whichwill determine how many consecutive steps are taken in the en+1 direction betweensteps in other directions. Each time that j steps are taken in the en+1 direction,all the ck, 1 ≤ k ≤ n, are incremented by 1; and each time that a step is taken in adirection other than en+1, cn+1 is incremented by 1. Therefore,(2.4)Ac(1, . . . , 1, in+1)=∑ξ∈Sn∑j1+···+jn+1=in+1cj1n+1(cξ(1) + j1)(cn+1 + 1)j2(cξ(2) + j1 + j2) · · ·· · · (cξ(n) + j1 + · · ·+ jn)(cn+1 + n)jn+1=∑ξ∈Sn∑j1+···+jn+1=in+1[n∏k=1(cn+1 + k − 1)jk(cξ(k) + j1 + · · ·+ jk)](cn+1 + n)jn+1= (cn+1 + n)in+1∑ξ∈Sn∑j1+···+jn+1=in+1n∏k=1[(cn+1 + k − 1)cn+1 + n]jk(cξ(k) + j1 + · · ·+ jk)= (cn+1 + n)in+1∑ξ∈Snin+1∑js≥0j1+···+jn=0n∏k=1[(cn+1 + k − 1)cn+1 + n]jk(cξ(k) + j1 + · · ·+ jk).This reasoning gives Formula (2.3). Clearly, Bc(u) is a convergent sum. 4 K. PETERSEN AND A. VARCHENKOSimilarly, existence of the limit in Formula (2.2) for any (i1, . . . , in) can be shownas follows. Let m, F , be the same as in Section 1. Then(2.5)Ac(i1, . . . , in, in+1) ==∑f∈F∑j1+···+jm+1=in+1cj1n+1(cf(1) + j1)(cn+1 + 1)j2(cf(2) + j1 + j2) · · ·· · · (cf(m) + j1 + · · ·+ jm)(cn+1 +m)jm+1= (cn+1 +m)in+1∑f∈Fin+1∑js≥0j1+···+jm=0m∏k=1[(cn+1 + k − 1)cn+1 +m]jk(cf(k) + j1 + · · ·+ jk).Dividing by (cn+1+m)in+1 and letting in+1 → ∞ again leaves a sum of convergentseries. Thus we get(2.6) Bc(i1, . . . , in) =∑f∈F∞∑j1,...,jn=0n∏k=1(cf(k) + j1 + · · ·+ jk)(cn+1 + k − 1cn+1 + n)jk.We shall give two formulas for Bc(u). In this section we evaluate Bc(u) usinga two-dimensional formula from [8]. In Section 3.1 another formula is obtainedby using derivatives of geometric series. Comparing the two formulas we get newidentities involving Stirling numbers of the first and second kinds (Corollary 3.11).Theorem 2.2. We have(2.7) Bc(u) =(cn+1 + n)nn!∑ξ∈Snn∏k=1(cξ(k) + cn+1 + k − 1) .2.2. Proof of Theorem 2.2. Denote by σi(c), i = 1, . . . , n, the elementary sym-metric functions in c,(2.8) σi(c) =∑1≤j1<···<ji≤ncj1 . . . cji .Denote σ0(c) = 1. Clearly, the function Bc(u) is a symmetric polynomial in c ofthe form(2.9)Bc(u) =n∑i=0i!(n− i)! σi(c)αi,n(cn+1) = n!σn(c)αn,n(cn+1)+(n− 1)!σn−1(c)αn−1,n(cn+1) + . . . + n!σ0(c)α0,n(cn+1) ,where αi,n(cn+1), i = 0, . . . , n, are suitable functions in cn+1. For example,(2.10)α0,n(cn+1) =∞∑j1,...,jn=0n∏k=1(j1 + · · ·+ jk)(cn+1 + k − 1cn+1 + n)jk,αn,n(cn+1) =∞∑j1,...,jn=0n∏k=1(cn+1 + k − 1cn+1 + n)jk,and so on.PATH COUNTS AND STIRLING NUMBERS 5Example 2.3. For n = 3,(2.11)B(c1,c2,c3)(u) =∞∑j1,j2=0[(c1 + j1)(c2 + j1 + j2) + (c2 + j1)(c1 + j1 + j2)] ××(c3c3 + 2)j1 (c3 + 1c3 + 2)j2= 2! c1c2 α2,2(c3) + (c1 + c2)α1,2(c3) + 2!α0,2(c3),where(2.12)α2,2(c3) =∞∑j1,j2=0(c3c3 + 2)j1 (c3 + 1c3 + 2)j2,α1,2(c3) =∞∑j1,j2=0(2j1 + j2)(c3c3 + 2)j1 (c3 + 1c3 + 2)j2,α0,2(c3) =∞∑j1,j2=0j1(j1 + j2)(c3c3 + 2)j1 (c3 + 1c3 + 2)j2.Our goal is to calculate all the coefficients αi,n(cn+1) and show that the resultgives Formula (2.7). To do so let us calculate the number(2.13) Bc1,c2(n) =∞∑j1,...,jn=0n∏k=1(c1 + j1 + · · ·+ jk)(c2 + k − 1c2 + n)jk.The function Bc1,c2(n) has the form:(2.14) Bc1,c2(n) = cn1αn,n(c2) + cn−11 αn−1,n(c2) + · · ·+ α0,n(c2),where the functions αi,n, i = 0, . . . , n, are the same functions of one variable as in(2.9). (To see this, note the identical form of the two products in formulas (2.3)and (2.13), and note that the sum over the symmetric group is absorbed into thecoefficients that do not involve cn+1.)Example 2.4. We have(2.15)Bc1,c2(2) =∞∑j1,j2=0(c1 + j1)(c1 + j1 + j2)(c2c2 + 2)j1 (c2 + 1c2 + 2)j2= c21 α2,2(c2) + c1 α1,2(c2) + α0,2(c2),where the functions α0,2, α1,2, α2,2 are the same functions of one variable as in(2.12).To calculate the functions αi,n and prove Theorem 2.2 it is enough to calculateBc1,c2(n).Lemma 2.5. We have(2.16) Bc1,c2(n) =(c2 + n)nn!n∏k=1(c1 + c2 + k − 1) .6 K. PETERSEN AND A. VARCHENKOProof. Let us use formula (1.1) of [8] to evaluate(2.17) Bc1,c2(n) = limi→∞Ac1,c2(n, i)(c2 + n)i.Formula (1.1) for Ac1,c2(n, i) has n+ 1 summands and the leading summand is(2.18)(c1 + c2 + n− 1n)(c2 + n)n+i.This term after division by (c2+n)i gives the right hand side of (2.17). The lemmais proved. Lemma 2.5 gives Theorem 2.2.2.3. Formula for Bc(i). Formula (2.7) for Bc1,...,cn+1(1, . . . , 1) easily gives a for-mula for Bc1,...,cn+1(i1, . . . , in) for any (i1, . . . , in) .Proposition 2.6. Let m, F , be the same as in Section 1. Then(2.19) Bc1,...,cn+1(i1, . . . , in) =(cn+1 +m)mm!∑f∈Fm∏j=1(cf(j) + cn+1 + j − 1) .3. Stirling numbers and related polynomialsTheorem 2.2 has as corollaries interesting identities involving Stirling numbersof the first and second kinds.3.1. Theorem 2.2 in new notation. Denoteua =q − aq, Da = ua∂∂ua.Then Theorem 2.2 says that for arbitrary numbers c1, . . . , cn we have(3.1)(∑σ∈Snn∏a=1(cσ(a) +Da +Da+1 + · · ·+Dn))n∏b=111− ub=qnn!∑σ∈Snn∏a=1(cσ(a)+q−a).Example 3.1. For n = 2, Formula (3.1) says[(c1 +D2)(c2 +D2 +D1) + (c2 +D2)(c1 +D2 +D1)]11− u111− u2=q22[(c1 + q − 1)(c2 + q − 2) + (c2 + q − 1)(c1 + q − 2)] .PATH COUNTS AND STIRLING NUMBERS 7This statement is equivalent to the identities11− u111− u2=q22,(2D2 +D1)11− u111− u2=q22(q − 2 + q − 1) ,D2(D2 +D1)11− u111− u2=q22(q − 1)(q − 2) .3.2. Formula for (u∂u)k(1−u)−1. We make some observations that are useful forhandling expressions such as the left side of Formula (3.1).Let q be a variable, a a positive integer. Set(3.2) u =q − aq, ∂u =∂∂u.Let A(i, j) = A0,0(i, j) be the standard Eulerian numbers as in Section 1.Theorem 3.2. For any positive integer k we have(3.3) (u∂u)k(1− u)−1 = a−k−1∑i+j=k−1A(i, j) qi+1(q − a)j+1 .Example 3.3.(3.4)u∂u(1 − u)−1 = a−2 q(q − a) ,(u∂u)2(1 − u)−1 = a−3 (q2(q − a) + q(q − a)2) ,(u∂u)3(1 − u)−1 = a−4 (q3(q − a) + 4q2(q − a)2 + q(q − a)3) .Theorem 3.2 follows by induction from the following lemma.Lemma 3.4. We haveu∂∂u=q(q − a)a∂∂q.Proof of Theorem 3.2.q(q − a)a∂∂qa−k−1∑i+j=k−1A(i, j) qi+1(q − a)j+1= a−k−2∑i+j=k−1A(i, j) (i + 1) qi+1(q − a)j+2+ a−k−2∑i+j=k−1A(i, i) (j + 1) qi+2(q − a)j+1= a−k−2∑r+s=kA(r, s− 1) (r + 1) qr+1(q − a)s+1+ a−k−2∑r+s=kA(r − 1, s) (s+ 1) qr+1(q − a)s+1= a−k−2∑r+s=kA(r, s) qr+1(q − a)s+1 .8 K. PETERSEN AND A. VARCHENKO3.3. Γ-polynomials. We shall reformulate Theorem 3.2 in terms of a slight vari-ation of Eulerian polynomials (see [3, p. 244]), which we call Γ-polynomials.Let s2(k,m) denote Stirling numbers of the second kind, which count the numberof partitions of a k-element set into m nonempty subsets, so thats2(1, 1) = 1,s2(2, 1) = 1, s2(2, 2) = 1,s2(3, 1) = 1, s2(3, 2) = 3, s2(3, 3) = 1.We haves2(k,m) = s2(k − 1,m− 1) +ms2(k − 1,m) .We will be interested in the polynomialsΓk(q, n) =k∑i=1(−1)k−i s2(k, i) i! qi−1nk−i .Example 3.5.Γ1(q, n) = 1, Γ2(q, n) = 2q − n, Γ3(q, n) = 6q2 − 6qn+ n2.Theorem 3.6 (Frobenius, [3, p. 244]). We haveΓk(q, n) =∑i+j=k−1A(i, j) qi(q − n)j .3.4. ∆-polynomials. We will also need ∆-polynomials.Looking at the right side of Formula (3.1) suggests investigating related polyno-mials and Stirling numbers of the first kind.Stirling numbers of the first kind are the coefficients in the expansion(q)n =n∑k=0s1(n, k)qk,where (q)n is the falling factorial(q)n = q(q − 1)(q − 2) · · · (q − n+ 1).We haves1(n, k) =∑1≤i1<···<in−k≤n−1(−1)n−k i1i2 · · · in−k .For natural numbers 1 ≤ k < n, define a polynomial∆n,k(q) =∑1≤i1<···<in−k≤n−1(q − i1)(q − i2) · · · (q − in−k).Define ∆n,n(q) = 1. We have ∆n,1(q) = (q − 1) . . . (q − n+ 1) and∆n,k(q) =n−k∑m=0(k − 1 +mm)s1(n, k +m) qm .PATH COUNTS AND STIRLING NUMBERS 9Example 3.7.∆2,2(q) = 1, ∆2,1(q) = q − 1,∆3,3(q) = 1, ∆3,2(q) = 2q − 3, ∆3,1(q) = q2 − 3q + 2.3.5. Relation between Γ and ∆ polynomials.Example 3.8. We have the following identity:(2−1(21)Γ1(q,2) 02−2(22)Γ2(q,2) 2−1(11)Γ1(q,2))(∆2,2(q)∆2,1(q))=(∆2,2(q)∆2,1(q)).In other words, we have(21)Γ1(q, 2)∆2,2(q)2= ∆2,2(q) ,(22)Γ2(q, 2)∆2,2(q)22+(11)Γ1(q, 2)∆2,1(q)2= ∆2,1(q) .Example 3.9. We have the following identity3−1(31)Γ1(q,3) 0 03−2(32)Γ2(q,3) 3−1(21)Γ1(q,3) 03−3(33)Γ3(q,3) 3−2(22)Γ2(q,3) 3−1(11)Γ1(q,3)∆3,3(q)∆3,2(q)∆3,1(q) =∆3,3(q)∆3,2(q)∆3,1(q) .In other words, we have(31)Γ1(q, 3)∆3,3(q)3= ∆3,3(q) ,(32)Γ2(q, 3)∆3,3(q)32+(21)Γ1(q, 3)∆3,2(q)3= ∆3,2(q) ,(33)Γ3(q, 3)∆3,3(q)33+(22)Γ2(q, 3)∆3,2(q)32+(11)Γ1(q, 3)∆3,1(q)3= ∆3,1(q) .More generally, we have the following relation.Theorem 3.10. For any 1 ≤ k ≤ n we have(3.5) ∆n,k(q) =n−k∑m=0(k +m1 +m)Γ1+m(q, n)∆n,k+m(q)n1+m.Proof. Recall the notations ub and Db defined at the beginning of Section 3.1.Denote(3.6)πn =n∏b=111− ub, Ikn(q) =∑1≤i1<···<ik≤nk∏j=1(Dij +Dij+1 + · · ·+Dn) πn .For example,I1n(q) = (nDn + (n− 1)Dn−1 + · · ·+D1)πn.Denote σi(c), i = 1, . . . , n, the elementary symmetric functions in c = (c1, . . . , cn)as before.10 K. PETERSEN AND A. VARCHENKOThe left hand side of Formula (3.1) equalsn∑k=0k!(n− k)!σn−k(c)Ikn(q).The right hand side equalsqnn!n∑k=0k!(n− k)!σn−k(c)∆n+1,n−k+1(q) .By comparing coefficients of the symmetric polynomials in these two equations wegetIkn(q) =qnn!∆n+1,n−k+1(q) .We have∆n+1,n−k+1(q)−∆n,n−k(q) = (q − n)∆n,n−k+1(q)andIkn(q) =k∑j=0(n− jk − j)Ijn−1(q)((Dn)k−j 11− un).Hence(3.7) Ikn(q)− Ikn−1(q)11 − un=k−1∑j=0(n− jk − j)Ijn−1(q)((Dn)k−j 11− un).The left hand side of (3.7) isIkn(q)− Ikn−1(q)11 − un= Ikn(q)− Ikn−1(q)qn=qnn!∆n+1,n−k+1(q)−qn−1(n− 1)!∆n,n−k(q)qn=qnn!(∆n+1,n−k+1(q)−∆n,n−k(q))=qn(q − n)n!∆n,n−k+1(q).The right hand side of (3.7) equalsk−1∑j=0(n− jk − j)Ijn−1(q)((Dn)k−j 11− un)=k−1∑j=0(n− jk − j)qn−1(n− 1)!∆n,n−j(q)q(q − n)nk−j+1Γn−j(q, n).These two equations prove Theorem 3.10. Since ∆n,n(q) = 1 and Γ1(q, n) = 1 for all n, identity (3.5) allows us to expressthe polynomials ∆n,k(q) in terms of polynomials Γk(q, n) and vice versa — see theexamples above.PATH COUNTS AND STIRLING NUMBERS 113.6. Identities involving Stirling numbers. The conclusion of Theorem 3.10can be written in the following form:(3.8) ∆n,k(q) =1n− kn−k∑m=1(m+ km+ 1)1nmΓm+1(q, n)∆n,k+m(q), k = 1, . . . , n.Comparing coefficients of qr of the two sides yields identities involving the Stirlingnumbers s1(n, k) and s2(n, k) of the first and second kinds, indexed with 1 ≤ k ≤ n(and equalling 0 outside this range).Corollary 3.11. For 1 ≤ k ≤ n, 0 ≤ r ≤ k, we have(3.9)(r + n− k − 1r)s1(n, r + n− k) =k∑m=0(m+ n− km+ 1) r∑i=0(i + n− k +m− 1i)(−1)m+r−inr−i+1(r − i+ 1)! ×s2(m+ 1, r − i + 1)s1(n, i+ n− k +m).For r = 0, this sayss1(n, n− k) =n− kns1(n, n− k) +k∑m=1(m+ n− km+ 1)(−1)mns1(n, n− k +m),and hence(3.10) s1(n, n− k) =1kk∑m=1(m+ n− km+ 1)(−1)ms1(n, n− k +m), k = 1, . . . , n.This contains the known formulass1(n, n− 1) = −(n2)and s1(n, n− 2) =124n(n− 1)(n− 2)(3n− 1).References[1] S. Bailey, M. Keane, K. Petersen, and I. Salama, Ergodicity of the adic transformation onthe Euler graph, Math. Proc. Camb. Phil. Soc. 141 (2006), no. 2, 231-238.[2] L. Carlitz and R. Scoville, Generalized Eulerian numbers: combinatorial applications, J. ReineAngew. Math. 265 (1974), 110–137. MR0429575 (55 #2587)[3] L. Comtet, Advanced Combinatorics, enlarged, D. Reidel Publishing Co., Dordrecht, 1974.The Art of Finite and Infinite Expansions.[4] P. Flajolet, P. Dumas, and V. Puyhaubert, Some exactly solvable models of urn process the-ory, DMTCS Proceedings, Fourth Colloquium on Mathematics and Computer Science Algo-rithms, Trees, Combinatorics and Probabilities (2006), 59-118. http://www.dmtcs.org/dmtcs-ojs/index.php/proceedings/issue/view/84.[5] S. Frick and K. Petersen, Random permutations and unique fully supported ergodicity for theEuler adic transformation, Ann. Inst. Henri Poincaré Probab. Stat. 44 (2008), no. 5, 876–885(English, with English and French summaries). MR2453848 (2009h:37004)[6] , Reinforced random walks and adic transformations. to appear in J. Theor. Prob.[7] A. Gnedin and G. Olshanski, The boundary of the Eulerian number triangle, Mosc. Math. J.6 (2006), no. 3, 461–475. MR2274860 (2007j:05006)12 K. PETERSEN AND A. VARCHENKO[8] K. Petersen and A. Varchenko, The Euler adic dynamical system and path counts in theEuler graph. To appear in Tokyo J. Math.[9] N. Sloane, The On-Line Encyclopedia of Integer Sequences, http://oeis.org, 2010.[10] N. Sloane and S. Plouffe, The Encyclopedia of Integer Sequences, Academic Press Inc., SanDiego, CA, 1995. With a separately available computer disk. MR1327059 (96a:11001)Department of Mathematics, University of North Carolina at Chapel Hill, ChapelHill, NC 27599-3250, USAE-mail address: petersen@math.unc.eduDepartment of Mathematics, University of North Carolina at Chapel Hill, ChapelHill, NC 27599-3250, USAE-mail address: anv@math.unc.edu

Path count asymptotics and Stirling numbers

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Path count asymptotics and Stirling numbers

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