

A113227


Number of permutations avoiding the pattern 1234.


5



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



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.
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



KEYWORD

nonn


AUTHOR



STATUS

approved



