reviewed

approved

proposed

reviewed

editing

proposed

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 [math.CO], 2011.

Lara Pudwell, <a href="https://doi.org/10.37236/301">Enumeration Schemes for Permutations Avoiding Barred Patterns, </a href="http://www.combinatorics.org/">, Electronic J. Combinatorics</a>, , Vol. 17 (1), 2010, R29, 27pp.

approved

editing

_David Callan (callan(AT)stat.wisc.edu), _, Dec 02 2011

editing

approved

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

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.

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.

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.

Barred Patterns, <a href="http://www.combinatorics.org/">Electronic J. Combinatorics</a>, Vol. 17 (1), 2010, R29, 27pp.

agrees Agrees with A122993 through n=8 term.

proposed

editing

editing

proposed

allocated for David CallanNumber of permutations that avoid the barred pattern bar{1}43bar{5}2

1, 1, 2, 5, 14, 43, 145, 538, 2194, 9790, 47491, 248706, 1396799, 8363711, 53121000, 356309314, 2514395528, 18606000547, 143956459002, 1161612656187, 9753494344997, 85044912003502, 768659919235828, 7189553986402426, 69486510911410279, 693003419860404514

0,3

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.

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

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.

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".

Clear[a];

a[0] = a[1] = 1;

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

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

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

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

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

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

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

agrees with A122993 through n=8 term

allocated

nonn

David Callan (callan(AT)stat.wisc.edu), Dec 02 2011

approved

editing

allocated for David Callan

allocated

approved