Pattern-Avoiding Permutations

are three distinct patterns. The first is known as a classical pattern (dashes in all  − 1 slots); the third is also known as a consecutive pattern (no dashes in any slots). Some authors call ̃ a “generalized pattern” and use the word “pattern” exclusively for what we call “classical patterns”. Let  =  1 2 · · ·  be a permutation on {1 2     }, where  ≥ . We say that  contains ̃ if there exist 1 ≤ 1  2       ≤  such that • for each 1 ≤  ≤ − 1, if  is empty, then +1 =  + 1; • for all 1 ≤  ≤ , 1 ≤  ≤ , we have      if and only if   . The string  1 2 · · ·   is called an occurrence of ̃ in  . If  does not contain ̃, then we say  avoids ̃ or that  is ̃-avoiding. For example,