A Combinatorial Problem Connected with Differential Equations

Davenport, H.; Schinzel, Andrzej

doi:10.2307/2373068

Cited by 122 publications

(122 citation statements)

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“…All the functionslXkare non-decreasing,and convergeto infinity with their argument. The same holds for IX too, which grows more slowly than any of the IXk' Note thatlX(n):'24 for all n:'2A (4) which is a tower with 65536 2's, thus lX(n):'24 for all practical purpOses.…”

Section: Statement Of Main Resultsmentioning

confidence: 73%

“…Is the maximal total length of such a sequence linear in the number of nodes of the corresponding tree? (4) There is a similarity between the two problems studied here and in [16], in that they both involve path compressions on trees and they both attain similar upper and lower bounds. Is there some general problem of this kind, of which both our problem and that of [16] are special instances?…”

Section: Open Problemsmentioning

confidence: 71%

“…(n) has been studied in several papers [2], [3], [4], [12], [15]. It is known (and easy to prove) that Al(n)=n and A2(n)=2n-1.…”

Section: Im)mentioning

confidence: 99%

“…Section 3 reviews Ithe definitions of Ackermann's function and its functional inverse, and states the main theorems of this paper. Section 4 establishes a linear equivalence between DS (n, 3) sequences and path compression schemes, so that the problem Can be reduced to that of estimating the maximal length of such schemes. Sections 5 and 6 contain the derivations of the upper and lower bounds, respectively.…”

Section: (N)=b(g(n») If Both F(n)= =O(g(n») and F(n)=q(g(n»)mentioning

confidence: 99%

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Nonlinearity of davenport—Schinzel sequences and of generalized path compression schemes

1986

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Davenport-Schinzel sequences are sequences that do not contain forbidden subsequences of alternating symbols. They arise in the computation of the envelope of a set of functions. We show that the maximal length of a Davenport-Schinzel sequence composed of n symbols is 6(noc(n»), where t1.(n)is the functional inverse of Ackermann's function, and is thus very slowly increasing to infinity. This is achieved by establishing an equivalence between such sequences and generalized path compression schemes on rooted trees, and then by analyzing these schemes.

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Section: Statement Of Main Resultsmentioning

confidence: 73%

Section: Open Problemsmentioning

confidence: 71%

“…(n) has been studied in several papers [2], [3], [4], [12], [15]. It is known (and easy to prove) that Al(n)=n and A2(n)=2n-1.…”

Section: Im)mentioning

confidence: 99%

Section: (N)=b(g(n») If Both F(n)= =O(g(n») and F(n)=q(g(n»)mentioning

confidence: 99%

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Nonlinearity of davenport—Schinzel sequences and of generalized path compression schemes

1986

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“…It can be seen that P is 3-noncrossing in Klazar's sense if and only if there do not exist 6 elements Klazar's definition of 3-noncrossing partitions is different from ours. For example, let P be the partition 15-246-37 of [7], with standard representation as follows: r r r r r r r 1 2 3 4 5 6 7…”

Section: Theorem 11 Asserts the Symmetric Distribution Of Cr(p ) Andmentioning

confidence: 99%

Crossings and nestings of matchings and partitions

Chen

Deng

et al. 2006

Trans. Amer. Math. Soc.

185

418

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Abstract. We present results on the enumeration of crossings and nestings for matchings and set partitions. Using a bijection between partitions and vacillating tableaux, we show that if we fix the sets of minimal block elements and maximal block elements, the crossing number and the nesting number of partitions have a symmetric joint distribution. It follows that the crossing numbers and the nesting numbers are distributed symmetrically over all partitions of [n], as well as over all matchings on [2n]. As a corollary, the number of knoncrossing partitions is equal to the number of k-nonnesting partitions. The same is also true for matchings. An application is given to the enumeration of matchings with no k-crossing (or with no k-nesting).

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A Formal Calculus for the Enumerative System of Sequences—I. Combinatorial Theorems

Jackson

Goulden

1979

Stud Appl Math

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The enumeration of sequences plays an important part in combinatorial enumeration since sequences may be used as a means for encoding combinatorial structures. In the main, the enumeration of sequences has been treated by a collection of methods individually applied in particular cases. We present here a uniform constructive method for dealing with a large class of sequence problems. We obtain the generating function as the trace of an expression involving certain incidence matrices, expressions which are obtainable immediately from the combinatorial structure of a problem. The method relies on certain properties of matrices whose rank is equal to one. We demonstrate that these properties may be used to derive the required generating functions explicitly or, in more complex cases, as solutions to systems of linear equations. Results for permutations may be derived from those for sequences, and in certain frequently encountered special cases this operation may be carried out by a straightforward substitution. Applications of the method presented here will be given in Part II.

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A Combinatorial Problem Connected with Differential Equations

Cited by 122 publications

References 0 publications

Nonlinearity of davenport—Schinzel sequences and of generalized path compression schemes

Nonlinearity of davenport—Schinzel sequences and of generalized path compression schemes

Crossings and nestings of matchings and partitions

A Formal Calculus for the Enumerative System of Sequences—I. Combinatorial Theorems

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