The following is a consequence of the isomorphism laid out in Section 1.5. Theorem 2.8.1. The closed class of permutations C, defined as Av(α), where α is any single permutation from the set {231,213,132,312} has the following properties:
α
Figure 2.8: The direct sum of α and β.
• C is atomic.
• C is enumerated by the Catalan numbers.
• C is partially well ordered.
Proof. The proof follows immediately from the isomorphism operations, 312 is the inverse of 231, 132 is the reverse of 231 and 213 is the complement of 231.
Theorem 2.8.2. The closed class of permutationsDdefined asAv(β)where
β is either321or123is also in one-to-one correspondence with stack sortable permutations.
Proof. It is clear that 321 is the reverse of 123 so we consider only a single case. We present the standard bijection (see B´ona [21, Lemma 4.3]) which fixes left to right minima and yields a slightly stronger result.
Definition 2.8.3. An element iof a permutation σ is a left-to-right mini- mum if there is no element smaller than iwhich lies to the left of it in the
permutation. Left-to-right maxima, right-to-left minima and right-to-left maxima are defined mutatis mutandis.
Letσ be a permutation which avoids 123. Label the left-to-right minima of σ as (m1, m2, . . . , mk). Let S denote the sequence of elements of σ which
are not left to right minima. It is immediately clear thatS is a decreasing sequence. Form a new permutation τ by keeping the left to right minima (m1, m2, . . . , mk) fixed and placing the elements of S from left to right at
each step placing the smallest element which has not yet been placed but which is larger than the closest left to right minima on the left, see figure 2.9. We see immediately thatτ is 132 avoiding and that τ is the only 132 avoider with this choice of left to right minima. Thus we have a bijection.
Figure 2.9: 123 and 132 avoiding permutations which are equivalent under our bijection
The proof of Theorem 2.8.2 extends naturally to a bijection between Av(σ⊕ 21) and Av(σ ⊕12), see Babson and West [17]; Backelin, West and Xin [18] extend the proof further to a bijection between Av(σ⊕(k . . .21)) and Av(σ⊕(12. . . k)).
Theorem 2.8.4 (see Atkinson, Murphy and Ruˇskuc [14]). The closed class of permutations D defined as Av(β) where β is either 321 or 123 is atomic.
235174. . .(n−2)(n−5)(n−1)(n)(n−3)} is called U, see Murphy [50]. Lemma 2.8.6. U is an infinite antichain.
Proof. Each member ofU contains a copy of the pattern 2341 as its leftmost four elements and as its topmost four elements, furthermore there are no other copies of 2341. Thus any embedding of one member into another will have to match these two parts. It is immediately clear that the chain like structure of the remaining elements makes such an embedding impossible, see figure 2.10. These end patterns are known generally as anchors, the central structure is, in this case, called an oscillation.
Figure 2.10: Two members of the antichain U.
Theorem 2.8.7 (see Murphy [50]). The closed class of permutationsD
Proof. To prove that a class is not partially well ordered we have merely to exhibit an infinite antichain inside it. It is clear that every member of U avoids 321 thus the class Av(321) is not partially well ordered.
The construction of just two more infinite antichains [50] is enough to yield the following:
Theorem 2.8.8.
There exists an algorithm which does the following:
Input: A closed class defined by a single basis element it avoids.
Output: TRUE if the class is partially well ordered, FALSE otherwise.
Proof. It is a result of Atkinson, Murphy and Ruˇskuc [14] that a closed class whose basis is a single permutationσ is partially well ordered if and only if σ∈ {1,12,21,132,213,231,312}.
A Token passing network is a directed graph with a specified input node, or source, and a specified output node, or sink. The remaining vertices have one of six types, each of which stores data in a different way. Tokens travel through the network along the edges, starting at the input and finishing at the output, only one token may move at any time. We assume the tokens enter the network in ascending order. The order in which the tokens leave the network is the permutation generated by the network.
Finite token passing networks, those that can hold only finitely many to- kens at any one time, have been studied at length. Such problems were first inspired by Knuth [45] who posed questions about systems of railway sid- ings. Atkinson and Tulley [11], Albert, Atkinson and Ruˇskuc [4] and finally Albert, Ruˇskuc and Linton [6] have all studied finite networks. We extend the concept, allowing networks to contain infinite components such as stacks and queues, in doing so we bring the work of Knuth [45], Pratt [53], Tarjan [59] and Atkinson, Murphy and Ruˇskuc [13] under the same definitional um- brella. We call such networks extended token passing networks. There have been many more variations of stack sorting and network sorting considered, for an overview see Bona [20].