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A372310 Number of permutations of length n avoiding the pattern 1324 and with 1 appearing before n. 0
1, 3, 11, 45, 198, 919, 4446, 22239, 114347, 601722, 3229614, 17632437, 97707195, 548538588, 3115293151, 17875151109, 103511938302, 604392787819, 3555410248782, 21057224371290, 125484804821226, 752020468811244, 4530163818778839, 27419805899781843, 166694596163875206 (list; graph; refs; listen; history; text; internal format)
OFFSET
2,2
COMMENTS
This sequence counts the number of permutations of size n written in one-line notation that avoid the pattern 1324 and have the 1 appearing before the n.
LINKS
Juan B. Gil, Oscar A. Lopez, and Michael D. Weiner, A positional statistic for 1324-avoiding permutations, arXiv:2311.18227 [math.CO], 2024.
FORMULA
G.f.: A(x) = (x*(B(x)-2))/(3-B(x)), where B(x) is the g.f. for A000139. (See arxiv paper by Gil, Lopez, Weiner.)
G.f. satisfies 0 = x^4*(8*x-1) + x^2*(9*x-1)*(4*x-1)*A(x) + x*(6*x-1)*(9*x-2)*A(x)^2 + (27*x^2-9*x+1)*A(x)^3.
EXAMPLE
For n=4, a(4)=11 is counting the permutations (in one-line notation): 1234, 1243, 1342, 1423, 1432, 2134, 2143, 2314, 3124, 3142, 3214.
MAPLE
f:= proc(n) f(n):= 2*(3*n)!/((2*n+1)!*(n+1)!) end:
a:= proc(n) option remember; `if`(n=1, 1,
add(a(n-i)*f(i), i=1..n))
end:
seq(a(n), n=2..26); # Alois P. Heinz, Apr 26 2024
CROSSREFS
Invert of shifted A000139.
Sequence in context: A151132 A200075 A187764 * A151133 A213333 A083886
KEYWORD
nonn
AUTHOR
Michael D. Weiner, Apr 26 2024
STATUS
approved

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Last modified June 16 04:47 EDT 2024. Contains 373420 sequences. (Running on oeis4.)