Some general results in combinatorial enumeration

Abstract

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

Growth rates of permutation classes: Categorization up to the uncountability threshold

“…In Section 8 we show that all sequences not eliminated by these considerations are realizable and prove Theorem 8.3. In Section 9 we prove Theorem 9.1 and exhibit a counterexample to a suggestion of Klazar [12]. We conclude in Section 10 with a discussion of the obstacles that would have to be overcome to extend our characterization.…”

Complexity Problems in Enumerative Combinatorics

“…The celebrated Marcus-Tardos theorem [121] states that {A n (ω)} is at most exponential, for all ω ∈ S k , with a large base of exponent for random ω ∈ S k [60]. We refer to [99,101,170] for many results on the subject, history and background.…”

Intervals of Permutation Class Growth Rates

“…The set of indecomposables in a permutation class in our family consists of Q 5,3 together with an element of F 5,3 n for each odd n 5 and an extra set from H. By Lemma 4.3, Q 5,3 contributes (q n ) = (1,1,2,3,5,7,8) to the enumeration of the indecomposables, and for each odd n 5, there are ten distinct generalised digits that enumerate sets of indecomposables in F 5,3 n , ranging between 1.1 and 1.2221. Let F n consist of this set of generalised digits for odd n 7 and So, by construction, for every permutation class S ∈ Φ 5,3,5,H there is a corresponding sequence (a n ), with each a n ∈ D n , that enumerates S. The minimal enumeration sequence is (ℓ n ) ≡ (1, 1, 2, 3,5,7,9,10,9) for which the growth rate is gr( (ℓ n )) ≈ 2.36028. Similarly, the maximal enumeration sequence is (u n ) ≡ (1, 1, 2, 3, 5, 7,9,11,13,14,13,12) for which the growth rate is gr( (u n )) ≈ 2.36420.…”

“…Let F n consist of this set of generalised digits for odd n 7 and So, by construction, for every permutation class S ∈ Φ 5,3,5,H there is a corresponding sequence (a n ), with each a n ∈ D n , that enumerates S. The minimal enumeration sequence is (ℓ n ) ≡ (1, 1, 2, 3,5,7,9,10,9) for which the growth rate is gr( (ℓ n )) ≈ 2.36028. Similarly, the maximal enumeration sequence is (u n ) ≡ (1, 1, 2, 3, 5, 7,9,11,13,14,13,12) for which the growth rate is gr( (u n )) ≈ 2.36420.…”