Catwalks, sandsteps and pascal pyramids

Journal of Integer Sequences

Volumn 3, Issue 1, 2000, Pages

  Guy, Richard K ^a  

^a UNIVERSITY OF CALGARY   (Canada)

Author keywords

[No Author keywords available]

Indexed keywords

INTEGER PROGRAMMING; MATHEMATICAL MODELS; NUMBER THEORY; PROBLEM SOLVING; SET THEORY; THEOREM PROVING;

CATALAN NUMBER; INTEGER SEQUENCES; LATTICE POINTS; PASCAL PYRAMIDS; SANDS PROBLEM;

COMBINATORIAL MATHEMATICS;

EID: 0141996692     PISSN: 15307638     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (80)

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10. n. Finally he discusses the connexion between these results, noncrossing partitions and Sperner's theorem. Ira Gessel]
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13. C2. Endre Csáki, Sri Gopal Mohanty & Jagdish Saran, On random walks in a plane, Ars Combin. 29 (1990) 309-318.
14. i of the first i members of a random sequence of n +1's and n-1's, where i = 1, 2,..., 2n, with limiting distribution in n → ∞ and the joint probability distribution for the number of waves mentioned above and the Galton statistic in the sequence with the limiting distribution. Then they give the joint probability distribution for the number of waves relative to the height k > 0 from the horizontal line and the length of time spent above this height expressed by the number of positive members in the well-defined sequence with the limiting distribution. They suggest the statistical tests based on these theorems in a two-sample problem. C. Hayashi (Tokyo)]
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18. d) for the number of lattice paths in the plane length d, with unit steps in the positive and negative coordinate directions, starting at the origin and ending at (a,b) which touch the line x = ky only at the initial point. In the present paper the authors note that this formula is correct if d = a+b, a-kb = 1, or k = 1; in other cases it is an upper bound. The authors use the method of cyclic permutation or "penetrating analysis" due to Dv. A method which yields an exact, but complicated, formula for this kind of problem was described in G1. Ira Gessel]
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21. n,m are the oldest ballot numbers. (For n
22. j are independent and assume the values ±1 with probability 1/2. The author derives for this symmetric random walk explicit formulas for the probability distribution of the number of returns to the origin, the number of changes of sign and other related quantities. The derivations are of a very elementary nature and the paper is self-contained. A more exhaustive treatment appears in Chapter III of the 2nd ed. of Fe (1957). J.L. Snell]
23. 2n-2k) is the number of paths of length 2n with exactly 2k steps lying above y = x. On p. 96 is given a bijection between paths from (0,0) to (k,k) and paths of length 2k which do not pass below y = x.]
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25. FI. Philippe Flajolȩt, Combinatorial aspects of continued fractions, Discrete Math., 32 (1980) 125-161; Ann. Discrete Math., 9 (1980) 217-222; MR 82f:05002ab. [Referred to in review of G3.]
26. nz) and are also given a combinatorial interpretation. They also include as a special case the q-Catalan numbers studied in Po. Ira Gessel]
27. Fv. Harry Furstenberg, Algebraic functions over finite fields, J. Algebra, 7 (1967) 271-277; MR 35 #6655. [Referred to in review of G1.]
28. G1. Ira M. Gessel, A factorization for formal Laurent series and lattice path enumeration, J. Combin. Theory Ser. A 28 (1980) 321-327; MR 81j:05012. [A powerful and striking factorization for certain formal Laurent series is proved, namely that the series is a product of a constant, a series in only negative powers and a series in only positive powers. Lagrange's formula for series reversion is treated as an application. Other applicationa are to the problems of enumerating restricted lattice paths (a novel interpretation of Laurent series in combinatorial theory) and to H. Furstenberg's theorem [Fv] that the diagonal of a rational power series in two variables is algebraic (giving a new formal method of showing that certain series are algebraic. D.G. Rogers]
29. G2. Ira M. Gessel, A probabilistic method for lattice path enumeration, J. Statist. Plann. Inference 14 (1986) 49-58; MR 87h:05017. [Some lattice path counting problems may be converted into problems of deriving distributions on random walks which give rise to functional equations. Solutions of these equations provide a probabilistic approach to the lattice path enumeration problems. The approach is illustrated by a few examples. Sri Gopal Mohanty]
30. G3. Ira M. Gessel, A combinatorial proof of the multivariable Lagrange inversion formula, J. Combin. Theory Ser. A 45 (1987) 178-195; MR 88h:05011. [Using the exponential generating function, the author gives a combinatorial proof of one form of the multivariable Lagrange inversion formula (MLIF). An outline of the proof is: (1) interpret the defining functional relations as generating functions for colored trees, (2) interpret the desired coefficient as the generating function for functions from a set to a larger set, (3) decompose the functional digraph from (2) into two types of connected components, whose generating functions give the MLIF. Labelle had given such a proof in one variable [L]. The author also gives a useful survey of forms of the MLIF given by Jacobi, Stieltjes, Good, Joni, and Abhyankar. He shows that Jacobi's form implies Good's form and gives a simple form generalizing that of Stieltjes, Joni, and Abhyankar. The paper includes some historical information on the Jacobi formula for matrices. det exp A = trace A. Dennis Stanton]
31. Go. Henry W. Gould, Final analysis of Vandermonde's convolution, Amer. Math. Monthly, 64 (1957) 409-415; MR 19, 379c. [Referred to in review of Ra.]
32. GJ. I.P. Goulden & D.M. Jackson, Path generating functions and continued fractions, J. Combin. Theory Ser. A, 41 (1986) 1-10; MR 871:05020.[The authors consider paths along the nonnegative integers in which each step consists of an increase of altitude of 1 (a rise), 0 (a level), or -1 (a fall). The paths are weighted to record the number of rises and levels at each altitude. The main result of the paper answers the following question: What is the sum of the weights of all paths with given initial and terminal altitudes, and with given bounds on maximum and minimum altitudes? The answer is expressed in terms of continued fractions and extends P. Flajolet's combinatorial theory of continued fractions [Fl]. Some classical identities for continued fractions are obtained as corollaries. Ira Gessel]
33. n-1. The paper also contains many references to work by physicists on problems of counting animals, which arise in studying thermodynamic models for critical phenomena and phase transitions. Ira Gessel]
34. jm+jn) and the sum over all partitions of k. He also gives a short introduction to enumerations in three-dimensional and derives a solution to a corresponding election problem, noting its agreement with MacMahon's. J. Riordan]
35. H1. B.R. Handa & Sri Gopal Mohanty, Enumeration of higher-dimensional paths under restrictions, Discrete Math. 26 (1979) 119-128; MR 81b:05012. [The authors consider the problem of counting lattice paths in k-dimensional space under restrictions. They obtain k-dimensional analogs of the familiar results in two dimensions. D.P. Roselle]
36. H2. B.R. Handa & Sri Gopal Mohanty, On a property of lattice paths, J. Statist. Plann. Inference 14 (1986) 59-62; MR 871:05023. [The authors give an algebraic discussion of the implications on lattice paths of the following fact: increasing sequences of integers such that the jth is at least b(j) greater than the (j-l)st, and is at most a(j), are in one-to-one correspondence to increasing sequences such that the jth is at most a(j) minus the sum of the first j b(k)'s. D.J. Kleitman]
37. Frank Harary, Topological concepts in graph theory, in Harary and Beineke, A Seminar on Graph Theory, Holt, Reinhart & Winston, New York & London, 1967, pp.13-17.
38. Hi. Terrell L. Hill, Steady-state kinetics of a linear array of interlocking reactions, Statistical Mechanics & Statistical Methods in Theory and Applications (Rochester NY), Plenum, New York, 1977, pp. 521-577; MR 57 #15112. [Referred to in review of Sh; the MR reference gives no further information.]
39. J. André Joyal, Une théorie combinatoire des séries formelles, Adv. in Math., 42 (1981) 1-82; MR 84d:05025. ["We present a combinatorial theory of formal power series. The combinatorial interpretation of formal power series is based on the concept of species of structures. A categorical approach is used to fomulate it. A new proof of Cayley's formula for the number of labelled trees is given as well as a new combinatorial proof (due to G. Labelle) of Lagrange's inversion formula. Pólya's enumeration theory of isomorphism classes of structures is entirely renewed. Recursive methods for computing cycle index polynomials are described. A combinatorial version of the implicit function theorem is stated and proved. The paper ends with general considerations on the use of coalgebras in combinatories."] K. S. Karlin & G. McGregor, Coincidence probabilities, Pacific J. Math., 9 (1959) 1141-1164; MR 22 #5072. [Among Gessel's references, but may be marginal; cf. immediately preceding paper and review.]
40. T.P. Kirkman, On the k-partitions of the r-gon and r-ace, Phil. Trans. 147 (1857) 225.
41. n, # 19 of the "Erdös-Pósa" problems - see Mi.]
42. Christian Krattenthaler, Counting lattice paths with a linear boundary. I. Österreich. Akad. Wiss. Math.-Natur. Kl. Sitzungsber. II 198 (1989) 87-107.
43. K1. Christian Krattenthaler, Enumeration of lattice paths and generating functions for skew plane partitions, Manuscripta Math., 63 (1989) 129-155.
44. 2 = (p + q + 1)!(2p + 2q +1)!/(p + 1)!(2p + 1)!(q + 1)!(2q + 1)! S.G. Williamson]
45. Mike Kuchinski, Catalan Structures and Correspondences, M.Sc. thesis, West Virginia University, Morgantown WV 26506, 1977.
46. L. Gilbert Labelle, Une nouvelle démonstration combinatoire des formules d'inversion de Lagrange, Adv. in Math., 42 (1981) 217-247; MR 83e:05016. [The purpose of this paper is to examine connexions between two classical results - the Lagrange inversion formula for power series and Cayley's formula for the number of labelled trees on n vertices - in the light of the combinatorial theory of formal series recently presented by J and founded on the notion of "species of structures". Some combinatorial operations are defined over species and correspond to analytic operations over their generating functions: hence, properties of the operations over species yield identities for formal series. The authors introduce two canonical constructions which associate a new species - "arborescence R-enrichie" and "endofonction R-enrichie", respectively - to any given species R and proves some deep isomorphism results. These yield, as simple corollaries, some generalized versions of Cayley's formula and the Lagrange inversion formula. Andrea Brini]
47. L1. Jacques Labelle & Yeong-Nan Yeh, Dyck paths of knight moves, Discrete Appl. Math., 24 (1989) 213-221; MR 90g:05017. [This paper enumerates lattice paths from the origin to a point along the x-axis which do not go below the x-axis, where the allowable moves are knight moves from left to right. The resulting generating function satisfies a fourth-degree polynomial equation. This compares with so-called Dyck paths, where the allowable moves are one-step diagonal moves to the right. In this classical case the resulting generating function satisfies a quadratic polynomial equation, whose solution yields the Catalan number generating function. The authors apply their methods to paths with (r,s) knight moves, and obtain polynomial equations of higher degree. Dennis White]
48. L2. Jacques Labelle & Yeong-Nan Yeh, Generalized Dyck paths, Discrete Math., 82 (1990) 1-6.
49. n-2) 0 < p < n, 0 < q < n. S. Bays]
50. Li. N. Linial, A new derivation of the counting formula for Young tableaux, J. Combin. Theory Ser. A, 33 (1982) 340-342; MR 83m:05016. [The well-known hook length formula for the number of standard Young tableaux of a given shape can be written in determinantal form (Frobenius) [see, e.g., D.E. Knuth, The Art of Computer Programming, Vol. 3, pp. 60-63, Addison-Wesley, 1973]. A short proof of this result is given by observing that the expansion of the determinant yields an alternating sum of mutinomial coefficients which obviously satisfies the difference equation for the numbers in question, together with the initial conditions. Volker Strehl]
51. Ly. R.C. Lyness, Al Capone and the death ray, Math. Gaz., 25 (1941) 283-287.
52. Ma. Major P.A. MacMahon, Combinatory Analysis, Vol. I, Section III, Chapter V, Cambridge Univ. Press, Cambridge, 1915. [Referred to in review of Ze and perhaps G5.]
53. Mi. E.G. Milner, Louis Pósa - a mathematical prodigy, Nabla, Bull. Malayan Math. Soc., 7 (1960) 61-64; Solutions to the Erd•s-Pósa problems I, II, ibid., 107-112, 154-159. [Problem 19 was to prove the identity mentioned under Kl].
54. Mi. E.G. Milner, Louis Pósa - a mathematical prodigy, Nabla, Bull. Malayan Math. Soc., 7 (1960) 61-64; Solutions to the Erd•s-Pósa problems I, II, ibid., 107-112, 154-159. [Problem 19 was to prove the identity mentioned under Kl].
55. M1. Sri Gopal Mohanty, Lattice Path Counting and Applications. Probability and Mathematical Statistics. Academic Press, 1979, xi+185 pp. [Page 2 gives the reflexion principle, whereby equations (9), (11), (12) and the formula at the foot of p. 8 of the original paper all follow from equation (15).]
56. M2. Sri Gopal Mohanty, On some generalization of a restricted random walk, Studio Sci. Math. Hungar., 3 (1968) 225-241; MR 39 #1022. [The author considers the paths of a restricted random walk starting from the origin, which at each step moves either one unit to the right or μ (positive integer) units to the left, and reaches the point m - μn in m+n steps. Random walks are considered schematically by representing each movement of the particle to the right or to the left by a horizontal or a vertical unit so that the restricted random walk corresponds to the minimal lattice paths the particle describes from the origin to (m,n). Expressions are obtained for total numbers of distinct paths under certain further conditions as follows. For a given path, say C, of such a random walk the total number of paths is found which, after each step, do not lie to the left of the corresponding point of C, and which touch C in a prescribed way in exactly r of the last s left steps. Expressions are obtained also for the number of paths crossing r times (but not necessarily reaching) a point α ≥ 0, for the number of paths reaching α, r times, and concerning the joint distribution of the numbers of times and steps in the region to the right of α. The last is shown to lead to a result connected with a ballot theorem of L. Takács [T]. The author mentions also related results due to E. Csáki [C0], E. Csáki & I. Vincze [C3, C4], K. Sen [Se] and O. Engleberg [E]. C.J. Ridler-Rowe]
57. Athanasios Papoulis, A new method of inversion of the Laplace transform, Q. App. Math. 14 (1957) 405-414; MR 18, 602e. [Finds Legendre coefficients; done earlier by Widder, Duke Math. J., 1 (1935) 126-136; and by Shohat ibid., 6 (1940) 615-626; MR 2, 98.]
58. P. J. Peacock, On "Al Capone and the Death Ray", Note 1633 Math. Gaz., 26 (1942) 218-219. [See note under Ly.]
59. n = (1/(4n - 2)))(2n n). The proofs of the results stated will be presented in a subsequent paper. A.L. Whiteman]
60. i. and L = 1 if m = n = 0. He then derives some identities involving the numbers L, and uses them to prove a Lagrange inversion formula on formal power series and a convolution formula given by Go. Rimhak Ree]
61. Gerhard Ringel, Färbungsprobleme auf Flächen und Graphen, VEB Deutscher Verlag der Wissenschaften, Berlin, 1959.
62. k+1) holds. Defining C(n;t,q,p)=A(n;t,q,p,0,1) and K(n;t,q,p)=A(n;t,q,p,1,0), the author deduces recurrences for C and K from which two classic q-Catalan numbers defined by Ca follow by taking t = q = 1. Taking u= 1, v = 1 in (**) leads to a new recurrence involving E(n;t,q,p) = A(n;t,q,p,1,1) which defines generalized Eulerian numbers. A conjecture involving the q-Catalan numbers is posed at the end of Section 5. R.N. Kalia]
63. R2. John Riordan, Combinatorial Identities
64. Bill Sands, Problem 1517*, Crux Mathematicorum 16 #2 (Feb. 1990) 44.
65. i - τ) and the number of steps above the height r. Applying the results obtained, the author proves some known relations for the unrestricted case also. The proofs of the theorems are of a combinatorial character, some of them involving one-to-one correspondences of paths. The paper has some points in common with E. I. Vincze]
66. 2). The author first shows that if the integers are chosen randomly, and a random path from (0,0) to (a,b) with unit steps east and north is chosen, then the probability that the path passes beneath all the stones is 1/(z+1). The author next considers a model for enzyme kinetics closely related to that studied by Hi and by SZ. In these models the states are all 0-1 sequences of length M. Each 0 or 1 represents an enzyme, which is either reduced (1) or oxidized (0). The transition rules essentially allow a 01 subsequence to become 10. Using the result of the first part of the paper, the author computes the steady-state distribution under transition probabilities which are different from those used in the earlier papers. Ira Gessel]
67. α, α10β → α10β where all the transition rates are equal. The authors give a formula for the steady state probabilities that the rth component of the string is zero. Petr Kürka]
68. N.J.A. Sloane, The On-Line Encyclopedia of Integer Sequences.
69. Su. Robert A. Sulanke, A recurrence restricted by a diagonal condition: generalized Catalan numbers, Fibonacci Quart., 27 (1989) 33-46; MR 90c:05012. [This paper contains a proof via lattice paths of the Lagrange inversion theorem for ordinary generating functions. It also includes many examples of the Catalan-Motzkin-Schröder sequence variety. A translation from lattice paths to planar trees is given along with several planar tree examples. Basically this paper considers paths where each step is of the form (x,y) → (x + j, y + 1), j ∈ {0, 1, 2,...}. Such a path from (0,0) to (k,l) is good if after leaving (0,0) all points on the path must lie above the line y = μx, Z. The number of good paths from (0,0) to (kμ + d) [sic!] is shown to be d/(1 + kμ) of all paths between the same points. These paths are then weighted and generating functions are introduced, leading, eventually, to a combinatorial proof of the Lagrange inversion theorem. Other combinatorial proofs include those of Ra (ordinary generating functions), L (exponential generating functions) and G. Louis Shapiro]
70. Sv. Marta Sved, Math. Intelligencer [Cf. Kl, Mi and see Gessel correspondence.]
71. a +b)]. The results extend those of many authors, some of the more recent being Chung, Feller, Hodges, Gnedenko, Rvaöeva. J. Kiefer]
72. T2. Lajos Takács, The distribution of majority times in a ballot, Z. Wahrscheinlichkeitstheorie und verw. Gebiete, 2 (1963) 118-121; MR 28 #3490. [In this sequel to a previous paper [T1] further probabilities are calculated concerning the number of times one candidate leads over another during the successive stages of a ballot. The combinatorial proofs are based on lemmas which are relevant also to fluctuation theory, order statistics and the theory of queues. The author also gives a generalization to processes with independent increments. F.L. Spitzer]
73. a-1 du, where a = 3π/(10 - 3π). Ira Gessel]
74. H.M. Taylor & R.C. Rowe, Note on a geometrical theorem, Proc. London Math. Soc. 13 (1882) 102-106.
75. 2(n + 2)!.
76. j are used and there are n! terms in the resulting formula, with alternating signs, corresponding to the n! permutations of the coordinates generated by the reflexions in the hyperplanes. A similar proof was given by Ze; the authors' proof describes in more detail the successive reflexions applied to a path. Ira Gessel]
77. W2. Toshihiro Watanabe, On a determinant sequence in the lattice path counting, J. Math. Anal. Appl. 123 (1987) 401-414; MR 88g:05015. [The number of lattice paths connecting two given lattice points and staying between upper and lower boundaries can be expressed by a determinant involving binomial coefficients, due to G. Kreweras. The recursive nature of this problem leads to a system of difference equations, and the same type of solution (determinants involving polynomials of binomial type) applies to a much larger class of operator equations. The author makes a new approach by associating such determinants with random tableaux. He obtains determinant sequences which satisfy a convolution identity similar to sequences of binomial type. The theory is then applied to Hill's enzyme model, reproducing a result of Shapiro and Zeilberger. Heinrich Niederhausen] [cf. K2, Hi, SZ]
78. x + ∑k=1 r(μk + 1)nk). The n-dimensional analog is obtained from a very general setting, leading to a so-called "multinomial basic polynomial sequence". Heinrich Niederhausen]
79. W4. Toshihiro Watanabe, On the Littlewood-Richardson rule in terms of lattice path combinatorics, Proc. First Japan Conf. Graph Theory & Appl., Discrete Math. 72 (1988) 385-390; MR 90b:05010. [This paper gives a proof of the Littlewood-Richardson rule for multiplying Schur functions by using the characterization of Schur functions as collections of nonintersecting lattice paths. The proof is based on Robinson's recomposition rule for transforming non-lattice paths into lattice paths. Dennis White]
80. i+1 = -1. A related approach, using recurrences instead of reflexion, was recently given by N. Linial [Li]. Ira Gessel]