A334154 -id:A334154 - cp's OEIS frontend

A334155 a(n) is the number of length n decorated permutations avoiding the pattern 001.

1, 2, 5, 15, 57, 273, 1593, 10953, 86553, 771993, 7666713, 83871513, 1001957913, 12976997913, 181106559513, 2709277004313, 43247182412313, 733699248716313, 13182759232076313, 250070586344012313, 4994229502288460313, 104743211837530700313, 2301653725221036620313

Offset: 0

Jordan Weaver, Apr 16 2020

nonn

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

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

Cf. A334154.

a(n) = n! + sum(j=0, n-1, (j+1)*j!); \\ Michel Marcus, May 11 2020

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

cp's OEIS Frontend

A334155 a(n) is the number of length n decorated permutations avoiding the pattern 001.

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