International audience Suppose that p,q,r,s are non-negative integers with m=p+q+r+s. The class X... more International audience Suppose that p,q,r,s are non-negative integers with m=p+q+r+s. The class X(p,q,r,s) of permutations that contain no pattern of the form α β γ where |α |=r, |γ |=s and β is any arrangement of \1,2,\ldots,p\∪ \m-q+1, m-q+2, \ldots,m\ is considered. A recurrence relation to enumerate the permutations of X(p,q,r,s) is established. The method of proof also shows that X(p,q,r,s)=X(p,q,1,0)X(1,0,r,s) in the sense of permutational composition.\par 2000 MATHEMATICS SUBJECT CLASSIFICATION: 05A05
The structure of the three pattern classes defined by the sets of forbidden permutations $\{2143,... more The structure of the three pattern classes defined by the sets of forbidden permutations $\{2143, 4321\}$, $\{2143, 4312\}$ and $\{1324, 4312\}$ is determined using the machinery of monotone grid classes. This allows the permutations in these classes to be described in terms of simple diagrams and regular languages and, using this, the rational generating functions which enumerate these classes are determined.
It is shown that there are ${2n\choose n}-\sum_{m=0}^{n-1}2^{n-m-1}{2m\choose m}$ permutations wh... more It is shown that there are ${2n\choose n}-\sum_{m=0}^{n-1}2^{n-m-1}{2m\choose m}$ permutations which are the union of an increasing sequence and a decreasing sequence.
A complete structural description and enumeration is found for permutations that avoid both $1324... more A complete structural description and enumeration is found for permutations that avoid both $1324$ and $4231$.
A priority queue, a container data structure equipped with the operations insert and delete-minim... more A priority queue, a container data structure equipped with the operations insert and delete-minimum, can re-order its input in various ways, depending both on the input and on the sequence of operations used. If a given input $\sigma$ can produce a particular output $\tau$ then $(\sigma,\tau)$ is said to be an allowable pair. It is shown that allowable pairs on a fixed multiset are in one-to-one correspondence with certain k-way trees and, consequently, the allowable pairs can be enumerated. Algorithms are presented for determining the number of allowable pairs with a fixed input component, or with a fixed output component. Finally, generating functions are used to study the maximum number of output components with a fixed input component, and a symmetry result is derived.
Transactions of the American Mathematical Society, 2013
A geometric grid class consists of those permutations that can be drawn on a specified set of lin... more A geometric grid class consists of those permutations that can be drawn on a specified set of line segments of slope ± 1 \pm 1 arranged in a rectangular pattern governed by a matrix. Using a mixture of geometric and language theoretic methods, we prove that such classes are specified by finite sets of forbidden permutations, are partially well ordered, and have rational generating functions. Furthermore, we show that these properties are inherited by the subclasses (under permutation involvement) of such classes, and establish the basic lattice theoretic properties of the collection of all such subclasses.
International audience Suppose that p,q,r,s are non-negative integers with m=p+q+r+s. The class X... more International audience Suppose that p,q,r,s are non-negative integers with m=p+q+r+s. The class X(p,q,r,s) of permutations that contain no pattern of the form α β γ where |α |=r, |γ |=s and β is any arrangement of \1,2,\ldots,p\∪ \m-q+1, m-q+2, \ldots,m\ is considered. A recurrence relation to enumerate the permutations of X(p,q,r,s) is established. The method of proof also shows that X(p,q,r,s)=X(p,q,1,0)X(1,0,r,s) in the sense of permutational composition.\par 2000 MATHEMATICS SUBJECT CLASSIFICATION: 05A05
The structure of the three pattern classes defined by the sets of forbidden permutations $\{2143,... more The structure of the three pattern classes defined by the sets of forbidden permutations $\{2143, 4321\}$, $\{2143, 4312\}$ and $\{1324, 4312\}$ is determined using the machinery of monotone grid classes. This allows the permutations in these classes to be described in terms of simple diagrams and regular languages and, using this, the rational generating functions which enumerate these classes are determined.
It is shown that there are ${2n\choose n}-\sum_{m=0}^{n-1}2^{n-m-1}{2m\choose m}$ permutations wh... more It is shown that there are ${2n\choose n}-\sum_{m=0}^{n-1}2^{n-m-1}{2m\choose m}$ permutations which are the union of an increasing sequence and a decreasing sequence.
A complete structural description and enumeration is found for permutations that avoid both $1324... more A complete structural description and enumeration is found for permutations that avoid both $1324$ and $4231$.
A priority queue, a container data structure equipped with the operations insert and delete-minim... more A priority queue, a container data structure equipped with the operations insert and delete-minimum, can re-order its input in various ways, depending both on the input and on the sequence of operations used. If a given input $\sigma$ can produce a particular output $\tau$ then $(\sigma,\tau)$ is said to be an allowable pair. It is shown that allowable pairs on a fixed multiset are in one-to-one correspondence with certain k-way trees and, consequently, the allowable pairs can be enumerated. Algorithms are presented for determining the number of allowable pairs with a fixed input component, or with a fixed output component. Finally, generating functions are used to study the maximum number of output components with a fixed input component, and a symmetry result is derived.
Transactions of the American Mathematical Society, 2013
A geometric grid class consists of those permutations that can be drawn on a specified set of lin... more A geometric grid class consists of those permutations that can be drawn on a specified set of line segments of slope ± 1 \pm 1 arranged in a rectangular pattern governed by a matrix. Using a mixture of geometric and language theoretic methods, we prove that such classes are specified by finite sets of forbidden permutations, are partially well ordered, and have rational generating functions. Furthermore, we show that these properties are inherited by the subclasses (under permutation involvement) of such classes, and establish the basic lattice theoretic properties of the collection of all such subclasses.
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Papers by Mike Atkinson