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a(n) is the number of length n decorated permutations avoiding the pattern 001.
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%I #21 May 18 2020 07:01:30

%S 1,2,5,15,57,273,1593,10953,86553,771993,7666713,83871513,1001957913,

%T 12976997913,181106559513,2709277004313,43247182412313,

%U 733699248716313,13182759232076313,250070586344012313,4994229502288460313,104743211837530700313,2301653725221036620313

%N a(n) is the number of length n decorated permutations avoiding the pattern 001.

%C A decorated permutation of length n is a word w=w_1...w_n on the letters {0,...,n} such that the restriction of w to its nonzero entries is an ordinary permutation in one-line notation. Then w avoids the pattern 001 if there is no subword w_{i_1}w_{i_2}w_{i_3} with i_1 < i_2 < i_3 such that w_{i_1}=w_{i_2} = 0 and w_{i_3} > 0.

%C a(n) is also the number of decorated permutations of length n avoiding the pattern 010. This can be proved via a simple bijection mapping a 001-avoiding decorated permutation to a 010-avoiding decorated permutation.

%C The number of decorated permutations of length n avoiding the pattern 012 is A334154.

%F a(n) = n! + Sum_{j=0..n-1} (j+1)*j!.

%e For n=3, the a(3)=15 decorated permutations avoiding 001 are 000, 010, 100, 012, 102, 120, 021, 201, 210, 123, 132, 213, 231, 312, and 321.

%e For n=5, 10302 does not avoid 001, because it contains the subword 002.

%o (PARI) a(n) = n! + sum(j=0, n-1, (j+1)*j!); \\ _Michel Marcus_, May 11 2020

%Y Cf. A334154.

%K nonn

%O 0,2

%A _Jordan Weaver_, Apr 16 2020