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Summary: On the Number of Permutations Avoiding a Given
Pattern
Noga Alon
Ehud Friedgut
February 22, 2002
Abstract
Let Sk and Sn be permutations. We say contains if there exist
1 x1 < x2 < . . . < xk n such that (xi) < (xj) if and only if (i) < (j). If
does not contain we say avoids .
Let F(n, ) = |{ Sn| avoids }|. Stanley and Wilf conjectured that for any Sk
there exists a constant c = c() such that F(n, ) cn for all n. Here we prove the
following weaker statement: For every fixed Sk, F(n, ) cn(n), where c = c()
and (n) is an extremely slow growing function, related to the Ackermann hierarchy.
1 Introduction
Let Sk and Sn be permutations. We say contains , and denote this by < , if
there exist 1 x1 < x2 < . . . < xk n such that (xi) < (xj) if and only if (i) < (j).
If does not contain we say avoids . Thus, (representing by (1), (2), . . . , (k))
1523647 contains 132 but avoids 321. Let
F(n, ) = |{ Sn| avoids }|.
For any S3 it is known (see, e.g., [9]) that F(n, ) = 2n
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