## Lattice Paths between Diagonal Boundaries (1998)

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Venue: | Electronic J. Combinatorics |

Citations: | 7 - 0 self |

### BibTeX

@ARTICLE{Niederhausen98latticepaths,

author = {Heinrich Niederhausen},

title = {Lattice Paths between Diagonal Boundaries},

journal = {Electronic J. Combinatorics},

year = {1998},

volume = {5},

pages = {30}

}

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### Abstract

A bivariate symmetric backwards recursion is of the form d[m, n]=w 0 (d[m1, n]+d[m, n-1])+# 1 (d[m-r 1 ,n-s 1 ]+d[m-s 1 ,n-r 1 ])++# k (d[m-r k ,n-s k ] +d[m-s k ,n-r k ]) where # 0 ,...# k are weights, r 1 ,...r k and s 1 ,...s k are positive integers. We prove three theorems about solving symmetric backwards recursions restricted to the diagonal band x + u<y<x-l. With a solution we mean a formula that expresses d[m, n] as a sum of di#erences of recursions without the band restriction. Depending on the application, the boundary conditions can take di#erent forms. The three theorems solve the following cases: d[x+u, x] = 0 for all x # 0, and d[x- l, x] = 0 for all x # l (applies to the exact distribution of the Kolmogorov-Smirnov two-sample statistic), d[x + u, x]=0 for all x # 0, and d[x - l +1,x]=d[x-l+1,x-1] for x # l (ordinary lattice paths with weighted left turns), and d[y, y - u +1]=d[y-1,y-u+1]for all y # u and d[x - l +1,x]=d[x-l+1,x-1] for x # l. The first theorem is a gene...

### Citations

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Citation Context ...s much later in a paper by Koroljuk [5] in 1955 (centered band), and Fray and Roselle [3] in 1971 (general case). S. G. Mohanty’s book is a general reference to Lattice Path Counting and Application=-=s [9]. -=-C. Jordan’s book Calculus of Finite Differences [4] is a classical reference to the power of differencing. A survey on the enumeration of lattice paths with weighted turns can be found in [6], which... |

82 |
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Citation Context ...ered band), and Fray and Roselle [3] in 1971 (general case). S. G. Mohanty’s book is a general reference to Lattice Path Counting and Applications [9]. C. Jordan’s book Calculus of Finite Differen=-=ces [4]-=- is a classical reference to the power of differencing. A survey on the enumeration of lattice paths with weighted turns can be found in [6], which contains proofs based on two-row arrays for (14) and... |

23 | On the foundations of combinatorial theory. VIII. Finite operator calculus - Rota, Kahaner, et al. - 1973 |

13 | The enumeration of lattice paths with respect to their number of turns
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- 1997
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Citation Context ...ations [9]. C. Jordan’s book Calculus of Finite Differences [4] is a classical reference to the power of differencing. A survey on the enumeration of lattice paths with weighted turns can be found i=-=n [6]-=-, which contains proofs based on two-row arrays for (14) and (16). For polynomial enumeration by turns of lattice paths above or below non-diagonal lines see [10]. I want to thank the referee for the ... |

12 |
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Citation Context ...ralized ballot path with steps →, ↑ and ↗ (king moves) is also called a Schröder path [14], [16]. The number of ordinary paths (10) above (or below) a diagonal boundary is usually attributed to=-= André [1]. -=-Takács [17] gives an excellent review of the history of lattice path counting. André does not use the Reflection Principle in his paper. The principle appears to be much older, and is sometimes refe... |

9 |
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Citation Context ...tribution of the Kolmogorov-Smirnov test. The exact number of ordinary paths between diagonal bounds, (21), appears much later in a paper by Koroljuk [5] in 1955 (centered band), and Fray and Roselle =-=[3] in -=-1971 (general case). S. G. Mohanty’s book is a general reference to Lattice Path Counting and Applications [9]. C. Jordan’s book Calculus of Finite Differences [4] is a classical reference to the ... |

4 |
Lattice Path Enumeration and Umbral Calculus
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Citation Context ...A formula for the entries in the left table can be found by elementary combinatorial arguments. The polynomials in the right table are easily constructed by algebraic methods like the Umbral Calculus =-=[11], d[m, n]=dn(m)= l=0 m-=-↑ n� � �� � m − n n − 2l + m n − 2l + m . n − 2l + m l n − l This paper is about non-polynomials tables. In the last two sections, however, we need the assumptions that the colum... |

3 |
Lattice paths with weighted left turns above a parallel to the diagonal
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(Show Context)
Citation Context ...∇2t[n−l+1,m−1+l] =t[m, n] −∇ −1 1 ∇2∇1∇2t[n−l+2,m+l]/µ = t[m, n] −∇ 2 2t[n−l+2,m+l]/µ. d[m, n]= � d≥0 µ d t[m, n]= � d≥0 �� �� � m n d d � �� � m=-= n d d Two other proofs of this result can be found in [7]. µ d � �� �-=- m − 2+l n − l +2 − d−1 d+1 + � . (16) Example 23 (vizier+knight+hook) The symmetric recursion ({(1, 2), (2, 1)} , (1,ν,ν)) is another subcase of the king+knight moves. Suppose, the a a a ... |

3 | Symmetric Sheffer sequences and their applications to lattice path counting, Journal of Statistical Planning and Inference 54 - Niederhausen - 1996 |

2 |
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Citation Context ...ontinuous version equivalent to the asymptotic distribution of the Kolmogorov-Smirnov test. The exact number of ordinary paths between diagonal bounds, (21), appears much later in a paper by Koroljuk =-=[5] in -=-1955 (centered band), and Fray and Roselle [3] in 1971 (general case). S. G. Mohanty’s book is a general reference to Lattice Path Counting and Applications [9]. C. Jordan’s book Calculus of Finit... |

2 | Einige Beispiele Brown’scher Molekularbewegung unter Einfluß äußerer Kräfte, Extrait du Bulletin de L’Académie des Sc. de Cracovie, Série A - Smoluchowski - 1913 |

1 |
A Guide to Fairy Chess
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Citation Context ...) (8) [K] [ a a a a a a a →, ↑, ↗, , * a a a a a king+knight unrestricted ˆ K] moves zeroes (19) zeroes zeroes 1.3 Notes There is no universally accepted catalog for naming lattice paths. Fairy=-= Chess [2] is -=-better regulated, and can serve as a reference for certain “unusual” moves. However, all step vectors in this paper have only nonnegative components, and therefore take only half of the possible m... |

1 |
Recursive initial values
- Niederhausen
- 1997
(Show Context)
Citation Context ...hs with weighted turns can be found in [6], which contains proofs based on two-row arrays for (14) and (16). For polynomial enumeration by turns of lattice paths above or below non-diagonal lines see =-=[10]-=-. I want to thank the referee for the careful reading of this paper, and the many insightful comments, which led to substantial improvements. 2 Recursive Boards Recursive boards are very simple mathem... |

1 |
A recurrence restricted by a diagonal condition, Fibonacci Q
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(Show Context)
Citation Context ...d ballot paths. A ballot path that ends on the diagonal is a Catalan path or (rotated) Dyck path [8]. A generalized ballot path with steps →, ↑ and ↗ (king moves) is also called a Schröder path=-= [14], [16]. T-=-he number of ordinary paths (10) above (or below) a diagonal boundary is usually attributed to André [1]. Takács [17] gives an excellent review of the history of lattice path counting. André does n... |

1 |
On the ballot theorems. Advances in Combinatorial Methods and Applications to Probability and Statistics
- Takács
- 1997
(Show Context)
Citation Context ...ot path with steps →, ↑ and ↗ (king moves) is also called a Schröder path [14], [16]. The number of ordinary paths (10) above (or below) a diagonal boundary is usually attributed to André [1].=-= Takács [17] gi-=-ves an excellent review of the history of lattice path counting. André does not use the Reflection Principle in his paper. The principle appears to be much older, and is sometimes referred to as d’... |