A113227 Number of permutations avoiding the pattern 1-23-4.

1, 1, 2, 6, 23, 105, 549, 3207, 20577, 143239, 1071704, 8555388, 72442465, 647479819, 6083742438, 59885558106, 615718710929, 6595077685263, 73424063891526, 847916751131054, 10138485386085013, 125310003360265231

Offset

0,3

Comments

a(n) is the number of permutations on [n] that avoid the mixed consecutive/scattered pattern 1-23-4 (also number that avoid 4-32-1).

From David Callan, Jul 25 2008: (Start)

a(n) appears to also count vertical-marked parallelogram polyominoes of perimeter 2n+2; vertical-marked 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..250

Juan S. Auli and Sergi Elizalde, Wilf equivalences between vincular patterns in inversion sequences, arXiv:2003.11533 [math.CO], 2020.

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, and Simone Rinaldi, Enumerating five families of pattern-avoiding inversion sequences; and introducing the powered Catalan numbers, arXiv:1808.04114 [math.CO], 2018.

David Callan, A bijection to count (1-23-4)-avoiding permutations, arXiv:1008.2375 [math.CO], 2010.

Matteo Cervetti, A generating tree with a single label for permutations avoiding the vincular pattern 1-32-4, arXiv:2103.00246 [math.CO], 2021.

Sylvie Corteel, Megan A. Martinez, Carla D. Savage, and 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, 138-155.

Steven Finch, Pattern-Avoiding Permutations [Broken link?]

Steven Finch, Pattern-Avoiding Permutations [Cached copy, with permission]

Zhicong Lin and Sherry H. F. Yan, Vincular patterns in inversion sequences, Applied Mathematics and Computation (2020), Vol. 364, 124672.

Zhicong Lin and Shishuo Fu, On 120-avoiding inversion and ascent sequences, arXiv:2003.11813 [math.CO], 2020.

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 3-letter pattern 1-23 and start with a; u[n, a, m, k] is the number of 1-23-4-avoiding permutations on [n] that start with a, have n in position k and for which m is the minimum of the first k-1 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). - Sergei N. Gladkovskii, Oct 10 2012

Conjecture: a(n) = R(n-1, 0) for n > 0 with a(0) = 1 where R(n, q) = R(n-1, q+1) + Sum_{j=0..q} (j+1)*R(n-1, j) for n > 0, q >= 0 with R(0, q) = 1 for q >= 0. - Mikhail Kurkov, Jan 05 2024

Example

12534 contains a scattered 1-2-3-4 pattern (1234 itself) but not a 1-23-4 because the 2 and 3 are not adjacent in the permutation.

Mathematica

v[n_, a_] := v[n, a] = Sum[StirlingS2[a-1, i-1]i^(n-a), {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[n-1];
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, n-k+1]}];
u[n_, a_, m_, k_]/; n>=3 && k==2 && a<n && m==a := u[n-1, a];
u[n_, a_, m_, k_]/; n>=3 && k>=3 && a<n && m==a := bi[n-a-1, k-2]v[k-1, 1]u[n-k+1, a];
u[n_, a_, m_, k_]/; n>=3 && k>=3 && a<n && m<=Min[a-1, n-k+1] := Sum[bi[n-a-1, j]bi[a-m-1, k-3-j]v[k-1, k-1-j]u[n-k+1, m], {j, Max[0, k-2-(a-m)], Min[n-a-1, k-3]}];
Table[u[n], {n, 0, 15}]

Crossrefs

Sequence in context: A137547 A137548 A080108 * A200406 A165489 A352367

Adjacent sequences: A113224 A113225 A113226 * A113228 A113229 A113230

Keyword

nonn,changed

Author

David Callan, Oct 19 2005

Status

approved