A201497 Number 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

Offset

0,3

Comments

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.

Links

Table of n, a(n) for n=0..25.

David Callan, A permutation pattern that illustrates the strong law of small numbers, arXiv:1111.6297

Lara Pudwell, Enumeration Schemes for Permutations Avoiding Barred Patterns, Electronic J. Combinatorics, Vol. 17 (1), 2010, R29, 27pp.

Example

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".

Mathematica

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}]

Crossrefs

Agrees with A122993 through n=8 term.

Sequence in context: A137551 A148333 A271270 * A122993 A137552 A137553

Adjacent sequences: A201494 A201495 A201496 * A201498 A201499 A201500

Keyword

nonn

Author

David Callan, Dec 02 2011

Status

approved