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A201497 Number of permutations that avoid the barred pattern bar{1}43bar{5}2. 0

%I

%S 1,1,2,5,14,43,145,538,2194,9790,47491,248706,1396799,8363711,

%T 53121000,356309314,2514395528,18606000547,143956459002,1161612656187,

%U 9753494344997,85044912003502,768659919235828,7189553986402426,69486510911410279,693003419860404514

%N Number of permutations that avoid the barred pattern bar{1}43bar{5}2.

%C a(n) is the number of permutations of [n] that avoid the barred pattern bar{1}43bar{5}2. A permutation p avoids bar{1}43bar{5}2 if each instance of a not-necessarily-consecutive 432 pattern in p is part of a 14352 pattern in p.

%H David Callan, <a href="http://arxiv.org/abs/1111.6297">A permutation pattern that illustrates the strong law of small numbers</a>, arXiv:1111.6297

%H Lara Pudwell, Enumeration Schemes for Permutations Avoiding Barred Patterns, <a href="http://www.combinatorics.org/">Electronic J. Combinatorics</a>, Vol. 17 (1), 2010, R29, 27pp.

%e 14352 is an avoider because the 432 has the required "1" and "5" in appropriate position, but 512463 is not because 543 is a 432 pattern with no available "1".

%t Clear[a];

%t a[0] = a[1] = 1;

%t a[n_] /; n >= 2 := BellB[n - 1] + 1 + 2^(n - 2) - n +

%t Sum[(Sum[Binomial[n - 4 - a + j - i, j - i] (i + 2)^b, {i, 0, j}] -

%t Binomial[n - 3 - a + j, j])*StirlingS2[a - b, j], {a, 0,

%t n - 3}, {b, 0, a - 1}, {j, 0, a - b}] +

%t Sum[Binomial[j + a + 1, j + 1] StirlingS2[n - 2 - a, j], {a, 0,

%t n - 2}, {j, 0, n - 2 - a}];

%t Table[a[n], {n, 0, 25}]

%Y Agrees with A122993 through n=8 term.

%K nonn

%O 0,3

%A _David Callan_, Dec 02 2011

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Last modified January 18 17:18 EST 2022. Contains 350455 sequences. (Running on oeis4.)