Using Noonan–Zeilberger Functional Equations to enumerate (in polynomial time!) generalized Wilf classes

Nakamura, Brian; Zeilberger, Doron

doi:10.1016/j.aam.2012.10.003

Cited by 10 publications

(23 citation statements)

References 15 publications

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“…For various patterns τ , functional equations were derived for enumerating permutations with r occurrences of τ in [10,11,12]. These functional equations were then used to derive enumeration algorithms.…”

Section: Enumerating With Functional Equationsmentioning

confidence: 99%

“…, 1) = f n (τ ; t) and that functional equations could be derived for the P n polynomial. The pattern τ = 123 was considered in [11,12], and the polynomial P n was defined to be:…”

Section: Functional Equations For Single Patternsmentioning

confidence: 99%

“…This paper is organized as follows. In Section 2, we review and outline the functional equations enumeration approach developed in [10,11]. In Section 3, we derive both rigorous results and empirical values for higher order moments and mixed moments for various random variables X σ .…”

Section: Introductionmentioning

confidence: 99%

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On the asymptotic statistics of the number of occurrences of multiple permutation patterns

Janson

Nakamura

Zeilberger

2015

Journal of Combinatorics

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We study statistical properties of the random variables Xσ(π), the number of occurrences of the pattern σ in the permutation π. We present two contrasting approaches to this problem: traditional probability theory and the "less traditional" computational approach. Through the perspective of the first one, we prove that for any pair of patterns σ and τ , the random variables Xσ and Xτ are jointly asymptotically normal (when the permutation is chosen from Sn). From the other perspective, we develop algorithms that can show asymptotic normality and joint asymptotic normality (up to a point) and derive explicit formulas for quite a few moments and mixed moments empirically, yet rigorously. The computational approach can also be extended to the case where permutations are drawn from a set of pattern avoiders to produce many empirical moments and mixed moments. This data suggests that some random variables are not asymptotically normal in this setting.

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Section: Enumerating With Functional Equationsmentioning

confidence: 99%

“…, 1) = f n (τ ; t) and that functional equations could be derived for the P n polynomial. The pattern τ = 123 was considered in [11,12], and the polynomial P n was defined to be:…”

Section: Functional Equations For Single Patternsmentioning

confidence: 99%

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On the asymptotic statistics of the number of occurrences of multiple permutation patterns

Janson

Nakamura

Zeilberger

2015

Journal of Combinatorics

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“…Although all the entries in the matrices are changed for consistency and notational convenience, we will continue to disregard the entries below the diagonal in subsequent matrices X k and the entries above the diagonal in subsequent matrices Y k . We can apply the same computational tricks shown in [18,17]. For example, it is not necessary to compute P n (t; X n , Y n ) completely symbolically and substitute x i,j = 1 and y i,j = 1 at the end.…”

Section: A General Approach To S N (1324 R)mentioning

confidence: 99%

Using functional equations to enumerate 1324-avoiding permutations

Johansson

Nakamura

2014

Advances in Applied Mathematics

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We consider the problem of enumerating permutations with exactly r occurrences of the pattern 1324 and derive functional equations for this general case as well as for the pattern avoidance (r = 0) case. The functional equations lead to a new algorithm for enumerating length n permutations that avoid 1324. This approach is used to enumerate the 1324-avoiders up to n = 31. We also extend those functional equations to account for the number of inversions and derive analogous algorithms.

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“…[146] is discussed by D. Zeilberger [151] and it gave rise to the notion of a Wilfian formula (also called a polynomial enumeration scheme), which is an algorithm working in time polynomial in n, for enumerative problems n → f (n) ∈ N 0 of the type Ω(n c ) = log(2 + f (n)) = O(n d ) (for some real constants 0 < c < d). It appears in the works on enumeration of Latin squares by D. S. Stones [140] or permutations with forbidden patterns by V. Vatter [144], B. Nakamura and D. Zeilberger [109], B. Nakamura [108] and others.…”

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confidence: 99%

Some general results in combinatorial enumeration

Klazar

2010

Permutation Patterns

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For enumerative problems, i.e. computable functions f : N → Z, we define the notion of an effective (or closed) formula. It is an algorithm computing f (n) in the number of steps that is polynomial in the combined size of the input n and the output f (n), both written in binary notation. We discuss many examples of enumerative problems for which such closed formulas are, or are not, known. These problems include (i) linear recurrence sequences and holonomic sequences, (ii) integer partitions, (iii) pattern-avoiding permutations, (iv) triangle-free graphs and (v) regular graphs. In part I we discuss problems (i) and (ii) and defer (iii)-(v) to part II. Besides other results, we prove here that every linear recurrence sequence of integers has an effective formula in our sense. Definition 1.1 (PIO formula). For a counting function f : N → Z, a PIO formula is an algorithm, called a PIO algorithm, that for some constants c, d ∈ N for every input n ∈ N computes the output f (n) ∈ Z in at most c · m(n) d = O(m(n) d ) = poly(m(n)) steps, where m(n) = m f (n) := log(1 + n) + log(2 + |f (n)|) measures the combined complexity of the input and the output. Similarly for counting function f : X → Z defined on a subset X ⊂ N.We think this is the precise and definitive notion of a "closed formula", and the yardstick one should use, possibly with some ramifications or weakenings, to determine if a solution to an enumerative problem is effective. We call counting functions possessing PIO formulas shortly PIO functions. Definition 1.1 repeats

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Using Noonan–Zeilberger Functional Equations to enumerate (in polynomial time!) generalized Wilf classes

Cited by 10 publications

References 15 publications

On the asymptotic statistics of the number of occurrences of multiple permutation patterns

On the asymptotic statistics of the number of occurrences of multiple permutation patterns

Using functional equations to enumerate 1324-avoiding permutations

Some general results in combinatorial enumeration

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