Counting permutations by alternating descents

We find the exponential generating function for permutations with all valleys even and all peaks odd, and use it to determine the asymptotics for its coefficients, answering a question posed by Liviu Nicolaescu. The generating function can be expressed as the reciprocal of a sum involving Euler numbers. We give two proofs of the formula. The first uses a system of differential equations. The second proof derives the generating function directly from general permutation enumeration techniques, using noncommutative symmetric functions. The generating function is an "alternating" analogue of David and Barton's generating function for permutations with no increasing runs of length 3 or more. Our general results give further alternating analogues of permutation enumeration formulas, including results of Chebikin and Remmel

Packing and Counting Permutations

A permutation class is a set of permutations closed under taking subpermutations. We study two aspects of permutation classes: enumeration and packing. Our work on enumeration consists of two campaigns. First, we enumerate all juxtaposition classes of the form “Av(abc) next to Av(xy)”, where abc and xy are permutations of lengths three and two, respectively. We represent elements from such a juxtaposition class by Dyck paths decorated with sequences of points. Context-free grammars are then used to enumerate these decorated Dyck paths. Second, we classify as algebraic the generating functions of 1×m permutation grid classes where one cell is context-free and the remaining cells are monotone. We rely on properties of combinatorial specifications of context-free classes and use operators to express juxtapositions. Repeated application of operators resolves cases for m > 2. We provide examples to re-prove known results and give new ones. Our methods are algorithmic and could be implemented on a PC. Our work on packing consolidates current knowledge about packing densities of 4-point permutations. We also improve the lower bounds for the packing densities of 1324 and 1342 and provide rigorous upper bounds for the packing densities of 1324, 1342, and 2413. All our bounds are within 10-4 of the true packing densities. Together with the known bounds, we have a fairly complete picture of 4-point packing densities. Additionally, we obtain several bounds (lower and upper) for permutations of length at least five. Our main tool for the upper bounds is the framework of flag algebras introduced by Razborov in 2007. We also present Permpack — a flag algebra package for permutations

Counting Permutations Modulo Pattern-Replacement Equivalences for Three-Letter Patterns

We study a family of equivalence relations on $S_n$, the group of permutations on $n$ letters, created in a manner similar to that of the Knuth relation and the forgotten relation. For our purposes, two permutations are in the same equivalence class if one can be reached from the other through a series of pattern-replacements using patterns whose order permutations are in the same part of a predetermined partition of $S_c$. When the partition is of $S_3$ and has one nontrivial part and that part is of size greater than two, we provide formulas for the number of classes created in each previously unsolved case. When the partition is of $S_3$ and has two nontrivial parts, each of size two (as do the Knuth and forgotten relations), we enumerate the classes for $13$ of the $14$ unresolved cases. In two of these cases, enumerations arise which are the same as those yielded by the Knuth and forgotten relations. The reasons for this phenomenon are still largely a mystery

Finding and Counting Permutations via CSPs

Permutation patterns and pattern avoidance have been intensively studied in combinatorics and computer science, going back at least to the seminal work of Knuth on stack-sorting (1968). Perhaps the most natural algorithmic question in this area is deciding whether a given permutation of length n contains a given pattern of length k. In this work we give two new algorithms for this well-studied problem, one whose running time is n^{k/4 + o(k)}, and a polynomial-space algorithm whose running time is the better of O(1.6181^n) and O(n^{k/2 + 1}). These results improve the earlier best bounds of n^{0.47k + o(k)} and O(1.79^n) due to Ahal and Rabinovich (2000) resp. Bruner and Lackner (2012) and are the fastest algorithms for the problem when k in Omega(log{n}). We show that both our new algorithms and the previous exponential-time algorithms in the literature can be viewed through the unifying lens of constraint-satisfaction. Our algorithms can also count, within the same running time, the number of occurrences of a pattern. We show that this result is close to optimal: solving the counting problem in time f(k) * n^{o(k/log{k})} would contradict the exponential-time hypothesis (ETH). For some special classes of patterns we obtain improved running times. We further prove that 3-increasing and 3-decreasing permutations can, in some sense, embed arbitrary permutations of almost linear length, which indicates that an algorithm with sub-exponential running time is unlikely, even for patterns from these restricted classes

Counting Permutations by Their Rigid Patterns

AbstractIn how many permutations does the patternτ occur exactly m times? In most cases, the answer is unknown. When we search for rigid patterns, on the other hand, we obtain exact formulas for the solution, in all cases considered

On counting permutations by pairs of congruence classes of major index

For a fixed positive integer n, let S_n denote the symmetric group of n! permutations on n symbols, and let maj(sigma) denote the major index of a permutation sigma. For positive integers k<m not greater than n and non-negative integers i and j, we give enumerative formulas for the cardinality of the set of permutations sigma in S_n with maj(sigma) congruent to i mod k and maj(sigma^(-1)) congruent to j mod m. When m divides n-1 and k divides n, we show that for all i,j, this cardinality equals (n!)/(km).Comment: 8 page