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# pp5 - 5 S EPARABILITY The direct sum operation introduced...

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5. S EPARABILITY The direct sum operation introduced in the Lecture 3 has an obvious sym- metry. The skew sum of the permutation π of length m and the permutation σ of length n is the permutation π σ defined by π σ i π i n for 1 i m , σ i m for m 1 i m n . Figure 5.1 shows an example. A permutation is called separable if it can be built from the permutation 1 with sums and skew sums. For example, the permutation 576984132 from Figure 5.1 is separable: 576984132 13254 4132 1 2143 1 132 1 21 21 1 1 21 1 1 1 1 1 1 1 1 1 . The separable permutations form a permutation class, as the reader is asked to verify in Exercise 1. 5.1. B ASIS AND E NUMERATION We have specified the separable permutations by describing their structure, so a natural first goal is to find the basis of this class. Theorem 5.1. The basis of the class of separable permutations consists of 2413 and 3142 . Proof. It is easy to check that 2413 and 3142 are not separable, so since the separable permutations form a class (Exercise 1), they are contained in Av 2413 , 3142 . Figure 5.1: An example of a skew sum: 13254 4132 57698 4132 . 23

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24 L ECTURE 5 S EPARABILITY π i 1 n π i 1 n π j Figure 5.2: Two situations which arise in the proof of Proposition 5.1. The gray regions represent areas where entries cannot lie. There is a gray region above π i because we chose π i as the largest entry to the left of the 1 ; similarly, there is a gray region to the right of π j because of the conditions on our choice of π j . Now we must show that every element of Av 2413 , 3142 is separa- ble. We prove this by induction on the length of the permutation. For n 3 the claim is trivial, as all such permutations are separable, so take π Av 2413 , 3142 of length n 4 and suppose that the claim is true for all shorter permutations. Clearly π is separable if and only if its reverse is sep- arable, so without loss of generality, we may assume that the entry 1 occurs before (to the left of) the entry n in π . Let π i denote the greatest entry to the left of the entry 1 . There cannot be an entry to the right of n which lies below π i
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pp5 - 5 S EPARABILITY The direct sum operation introduced...

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