By K.V.S. Sarma
Facts MADE easy : DO IT your self ON PC-PHI-SARMA, K.V.S.-2010-EDN-2
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Extra info for Statistics Made Simple: Do it Yourself on PC
It is intriguing that the patterns of a given length cannot necessarily be ordered according to avoidance: Stankova and West  observed that sn (53241) < sn (43251) for all n ≤ 12 while s13 (53241) > s13 (43251). However, Stankova and West conjectured (for classical patterns) that such ordering can be done asymptotically (a pattern τ1 is asymptotically more avoidable than a pattern τ2 if sn (τ1 ) > sn (τ2 ) for sufficiently large n). 2 is the following problem, which is attracting significant attention from many researchers, and is probably the best known unsolved question in the area.
What is a pattern? 5. If k = 0, then we get an occurrence of 213 and we are done. Assuming k > 0, consider the letter, say x, immediately to the right of a. If x > b then bax is an occurrence of 213; if x < b then bxc is an occurrence of the pattern 213 with k − 1 letters between x and c. In both cases, we get a contradiction with the assumption on minimality of k with the given properties. Thus, π contains 213 and we are done. 6. 6. () For any n ≥ 0, Sn (3142) = Sn (41352). Thus, Av(3142) = Av(41352).
3 Input-restricted and output-restricted deques An input-restricted deque, introduced by Knuth  is similar to a stack in that it has a push operation, but the pop operation can remove an element from either end of the deque. A successful sorting of a permutation requires the existence of a sequence involving the allowed operations that leads to the increasing permutation. Of course, we now have more possibilities to sort a permutation. For example, the reader may check that the permutation 2341 requires three stacks in series to 36 Chapter 2.
Statistics Made Simple: Do it Yourself on PC by K.V.S. Sarma