

A113227


Number of permutations avoiding the pattern 1234.


2



1, 1, 2, 6, 23, 105, 549, 3207, 20577, 143239, 1071704, 8555388, 72442465, 647479819, 6083742438, 59885558106, 615718710929, 6595077685263, 73424063891526, 847916751131054, 10138485386085013, 125310003360265231
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,3


COMMENTS

a(n) is the number of permutations on [n] that avoid the mixed consecutive/scattered pattern 1234 (also number that avoid 4321).
From David Callan, Jul 25 2008: (Start)
a(n) appears to also count verticalmarked parallelogram polyominoes of perimeter 2n+2; verticalmarked means that for each vertical line that splits the polyomino into two nonempty polyominoes one of the unit segments on the common boundary is marked.
....._
..._.
._...
_._._
For example, the polyomino above, with n=5, has two such vertical lines, the left line giving only one choice for marking and the right line giving two choices. (End)


LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..150
A. M. Baxter, Algorithms for Permutation Statistics, Ph. D. Dissertation, Rutgers University, May 2011.
Andrew M. Baxter and Lara K. Pudwell, Enumeration schemes for vincular patterns, arXiv preprint arXiv:1108.2642 [math.CO], 2011.
Nicholas R. Beaton, Mathilde Bouvel, Veronica Guerrini, Simone Rinaldi, Enumerating five families of patternavoiding inversion sequences; and introducing the powered Catalan numbers, arXiv:1808.04114 [math.CO], 2018.
David Callan, A bijection to count (1234)avoiding permutations, arXiv:1008.2375 [math.CO], 2010.
Sylvie Corteel, Megan A. Martinez, Carla D. Savage, Michael Weselcouch, Patterns in Inversion Sequences I, arXiv:1510.05434 [math.CO], 2015.
Sergi Elizalde, Asymptotic enumeration of permutations avoiding generalized patterns, arXiv:math/0505254 [math.CO], 2005.
Sergi Elizalde, Asymptotic enumeration of permutations avoiding generalized patterns, Adv. in Appl. Math. 36 (2006), no. 2, 138155.
Steven Finch, PatternAvoiding Permutations [Broken link?]
Steven Finch, PatternAvoiding Permutations [Cached copy, with permission]
Zhicong Lin, Sherry H. F. Yan, Vincular patterns in inversion sequences, Applied Mathematics and Computation (2020), Vol. 364, 124672.
Megan A. Martinez and Carla D. Savage, Patterns in Inversion Sequences II: Inversion Sequences Avoiding Triples of Relations, arXiv:1609.08106 [math.CO], 2016.


FORMULA

In the recurrence coded in Mathematica below, v[n, a] is the number of permutations on [n] that avoid the 3letter pattern 123 and start with a; u[n, a, m, k] is the number of 1234avoiding permutations on [n] that start with a, have n in position k and for which m is the minimum of the first k1 entries. In the last sum, j is the number of entries lying strictly between a and n both in value and position.
From Gary W. Adamson, Jul 08 2011: (Start)
a(n) = the upper left term in M^n, M = the production matrix:
1, 1
1, 2, 1
1, 2, 3, 1
1, 2, 3, 4, 1
1, 2, 3, 4, 5, 1
...
(End)
G.f.: 1+x/(U(0)x) where U(k)= 1  x*k  x/U(k+1) ; (continued fraction, 1step).  Sergei N. Gladkovskii, Oct 10 2012


EXAMPLE

12534 contains a scattered 1234 pattern (1234 itself) but not a 1234 because the 2 and 3 are not adjacent in the permutation.


MATHEMATICA

Clear[u, v, w]; v[n_, a_] := v[n, a] = Sum[StirlingS2[a1, i1]i^(na), {i, a}]; u[0]=u[1]=1; u[n_]/; n>=2 := u[n] = Sum[u[n, a], {a, n}]; u[1, 1]=u[2, 1]=u[2, 2]=1; u[n_, a_]/; n>=3 && a==n := u[n1]; u[n_, a_]/; n>=3 && a<n := u[n, a] = u[n, a, a, 2] + Sum[u[n, a, m, k], {k, 3, n}, {m, Min[a, nk+1]}]; u[n_, a_, m_, k_]/; n>=3 && k==2 && a<n && m==a := u[n1, a]; u[n_, a_, m_, k_]/; n>=3 && k>=3 && a<n && m==a := bi[na1, k2]v[k1, 1]u[nk+1, a]; u[n_, a_, m_, k_]/; n>=3 && k>=3 && a<n && m<=Min[a1, nk+1] := Sum[bi[na1, j]bi[am1, k3j]v[k1, k1j]u[nk+1, m], {j, Max[0, k2(am)], Min[na1, k3]}]; Table[u[n], {n, 0, 15}]


CROSSREFS

Sequence in context: A137547 A137548 A080108 * A200406 A165489 A192315
Adjacent sequences: A113224 A113225 A113226 * A113228 A113229 A113230


KEYWORD

nonn


AUTHOR

David Callan, Oct 19 2005


STATUS

approved



