

A111277


Number of permutations avoiding the patterns {2413,4213,2431,4231,4321}; also number of permutations avoiding the patterns {3142,3412,3421,4312,4321}; number of weak sorting class based on 2413 or 3142.


3



1, 1, 2, 6, 19, 59, 180, 544, 1637, 4917, 14758, 44282, 132855, 398575, 1195736, 3587220, 10761673, 32285033, 96855114, 290565358, 871696091, 2615088291, 7845264892, 23535794696, 70607384109, 211822152349, 635466457070
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OFFSET

0,3


COMMENTS

a(n) = number of permutation tableaux of size n (A000142) for which each row is constant (all 1's, all 0's, or empty). For example, a(4)=19 counts all 4! permutation tableaux of size 4 except the following five: {{0, 1}, {1}}, {{1, 1}, {0, 1}}, {{0, 1}, {1, 1}}, {{1, 0}, {0, 1}}, {{1, 0}, {1, 1}}.  David Callan, Oct 06 2006
a(n) = number of distinct excedance specifications taken over all permutations on [n]. The excedance specification for a permutation (p_1, p_2, ..., p_n) is the sequence (a_1, a_2, ..., a_n) defined by a_i = 1, 0, or 1 according as p_i is greater than, equal to, or less than i. If all permutations with a given excedance specification are arranged in lex (dictionary) order, then the firstand only the firstavoids the pattern set {3142,3412,3421,4312,4321}.  David Callan, Jul 25 2008
a(n) = number of (1,0,1)sequences of length n such that the first nonzero entry is 1 and the last nonzero entry is 1 because these sequences are the valid excedance specifications. Example: a(3)=6 counts (1,1,1), (1,0,1), (1,1,0), (1,1,1), (0,1,1), (0,0,0).  David Callan, Jul 25 2008
Inverse binomial transformation leads to 0,1,0,3,3,9,... (offset 0), essentially to A062510.  R. J. Mathar, Jun 25 2011
A128308 is defined as A007318 * A128307; since A007318 is the Riordan array (1/(1x), x/(1x)) and A128307 is the Riordan array ((1x)^2/(12x), x), the first column of A128308 has g.f. (12x)^2/((13x)(1x)^2), which coincides with the g.f. of this sequence.  Peter J. Taylor, Jul 24 2014
Number of permutations of length n>0 avoiding the partially ordered pattern (POP) {1>2, 1>3, 4>3} of length 4. That is, number of length n permutations having no subsequences of length 4 in which the first element is larger than the second and third elements, and the fourth element is larger than the third element.  Sergey Kitaev, Dec 09 2020


LINKS

M. Albert, R. Aldred, M. Atkinson, C. Handley, D. Holton, D. McCaughan and H. van Ditmarsch, Sorting Classes, Elec. J. of Comb. 12 (2005) R31


FORMULA

a(n) = (3^n2*n+3)/4.
a(n) = +5*a(n1) 7*a(n2) +3*a(n3).  R. J. Mathar, Jun 25 2011
a(n+1) = sum of row 1 terms of M^n, an infinite bidiagonal matrix with all 1's in the superdiagonal and (1,3,0,0,0,...) in the main diagonal, and the rest zeros. Example: a(5) = 59 = (sum of row 1 terms of M^4) = (1 + 40 + 13 + 4 + 1).  Gary W. Adamson, Jun 23 2011
G.f.: (12*x)^2/((13*x)*(1x)^2).  R. J. Mathar, Jun 25 2011


MAPLE

A111277 := proc(n) (3^n2*n+3)/4 ; end proc:


MATHEMATICA



PROG



CROSSREFS



KEYWORD

nonn,easy


AUTHOR



EXTENSIONS



STATUS

approved



