A193285 -id:A193285

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A193284

Number of allowed patterns of length n of the map f(x) = 4x(1-x) on the unit interval. A permutation pi is an allowed pattern if there exists x in [0,1] such that the values x,f(x),f(f(x)),...,f^{n-1}(x) are different and in the same relative order as pi_1,pi_2,...,pi_n.

+10
2

1, 1, 2, 5, 12, 31, 75, 178, 414, 949, 2137, 4767

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

a(n) is also the number of allowed patterns of length n of the tent map x -> 1-|1-2x| in [0,1].

LINKS

Table of n, a(n) for n=0..11.

S. Elizalde and Y. Liu, On basic forbidden patterns of functions, Discrete Appl. Math. 159 (2011), 1207-1216.

FORMULA

a(n) = n! - A193285(n).

EXAMPLE

a(3) = 5 because the allowed patterns of length 3 are 123, 132, 213, 231, 312.

CROSSREFS

Cf. A000142, A193285 (forbidden patterns).

KEYWORD

nonn,more

AUTHOR

Sergi Elizalde, Jul 20 2011

EXTENSIONS

a(0)=1 prepended by Alois P. Heinz, Mar 02 2020

STATUS

approved

A194341

Triangular array: g(n,k)=number of fractional parts (i*r) in interval [(k-1)/n, k/n], for 1<=i<=n, 1<=k<=n, r=3-e.

+10
2

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 1, 1, 1, 1, 1, 0, 1, 2, 1, 1, 1, 1, 1, 1, 0, 1, 2, 0, 1, 2, 1, 0, 2, 1, 0, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 2, 0, 2, 2, 0, 2, 0, 1, 1, 1, 1, 0, 2, 0, 2, 1, 1, 1, 1, 2, 0, 2, 0, 1, 0, 2, 0, 2, 0, 2, 0, 2, 0

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,13

COMMENTS

See A194285.

LINKS

Table of n, a(n) for n=1..100.

EXAMPLE

First ten rows:

1..1

1..1..1

1..1..1..1

1..1..2..0..1

1..1..1..1..1..1

1..1..1..1..1..1..1

0..1..2..1..1..1..1..1

0..1..2..0..1..2..1..0..2..1

MATHEMATICA

r = 3-E;

f[n_, k_, i_] := If[(k - 1)/n <= FractionalPart[i*r] < k/n, 1, 0]

g[n_, k_] := Sum[f[n, k, i], {i, 1, n}]

TableForm[Table[g[n, k], {n, 1, 14}, {k, 1, n}]]

Flatten[%] (* A194341 *)

CROSSREFS

Cf. A193285.

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Aug 22 2011

STATUS

approved

A193316

Number of basic forbidden patterns of length n of the map f(x)=4x(1-x) on the unit interval.

+10
0

0, 0, 1, 5, 9, 28, 53, 110, 229, 474

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

COMMENTS

A permutation pi is a forbidden pattern if there is no x in [0,1] such that the values x,f(x),f(f(x)),...,f^{n-1}(x) are in the same relative order as pi_1,pi_2,...,pi_n. A forbidden pattern is basic if it is minimally forbidden, that is, the patterns obtained by removing pi_1 or pi_n are not forbidden.

a(n) is also the number of basic forbidden patterns of length n of the tent map x -> 1-|1-2x| in [0,1].

LINKS

Table of n, a(n) for n=1..10.

S. Elizalde and Y. Liu, On basic forbidden patterns of functions, arXiv:0909.2277 [math.CO], 2009.

S. Elizalde and Y. Liu, On basic forbidden patterns of functions, Discrete Appl. Math. 159 (2011), 1207-1216.

EXAMPLE

a(3) = 1 because the only basic forbidden pattern of length 3 is 321.

a(4) = 5 because the basic forbidden patterns of length 4 are 1423, 2134, 2143, 3142, 4231.

CROSSREFS

Cf. A193284, A193285.

KEYWORD

nonn,more

AUTHOR

Sergi Elizalde, Jul 22 2011

STATUS

approved

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