A333833 -id:A333833 - cp's OEIS frontend

A363181 Number of permutations p of [n] such that for each i in [n] we have: (i>1) and |p(i)-p(i-1)| = 1 or (i

1, 0, 2, 2, 8, 14, 54, 128, 498, 1426, 5736, 18814, 78886, 287296, 1258018, 4986402, 22789000, 96966318, 461790998, 2088374592, 10343408786, 49343711666, 253644381032, 1268995609502, 6756470362374, 35285321738624, 194220286045506, 1054759508543554

Offset: 0

Alois P. Heinz, May 19 2023

nonn

Number of permutations p of [n] such that each element in p has at least one neighbor whose value is smaller or larger by one.

Number of permutations of [n] having n occurrences of the 1-box pattern.

			a(0) = 1: (), the empty permutation.
a(1) = 0.
a(2) = 2: 12, 21.
a(3) = 2: 123, 321.
a(4) = 8: 1234, 1243, 2134, 2143, 3412, 3421, 4312, 4321.
a(5) = 14: 12345, 12354, 12543, 21345, 21543, 32145, 32154, 34512, 34521, 45123, 45321, 54123, 54312, 54321.
a(6) = 54: 123456, 123465, 123654, 124356, 124365, 125634, 125643, 126534, 126543, 213456, 213465, 214356, 214365, 215634, 215643, 216534, 216543, 321456, 321654, 341256, 341265, 342156, 342165, 345612, 345621, 346512, 346521, 431256, 431265, 432156, 432165, 435612, 435621, 436512, 436521, 456123, 456321, 561234, 561243, 562134, 562143, 563412, 563421, 564312, 564321, 651234, 651243, 652134, 652143, 653412, 653421, 654123, 654312, 654321.

Alois P. Heinz, Table of n, a(n) for n = 0..803
FindStat, St000064: The number of one-box pattern of a permutation.
S. Kitaev, J. Remmel, The 1-box pattern on pattern avoiding permutations, arXiv:1305.6970 [math.CO], 2013.

Main diagonal of A346462.

Cf. A002464, A003274, A011655, A095816, A127697, A179957, A229730, A271212, A211694, A333833, A363236.

a:= proc(n) option remember; `if`(n<4, [1, 0, 2$2][n+1],
      3/2*a(n-1)+(n-3/2)*a(n-2)-(n-5/2)*a(n-3)+(n-4)*a(n-4))
    end:
seq(a(n), n=0..30);

a(n) = A346462(n,n).
a(n)/2 mod 2 = A011655(n-1) for n>=1.
a(n) ~ sqrt(Pi) * n^((n+1)/2) / (2 * exp(n/2 - sqrt(n)/2 + 7/16)) * (1 - 119/(192*sqrt(n))). - Vaclav Kotesovec, May 26 2023

A263690 Number of length n arrays of permutations of 0..n-1 with each element moved by -n to n places and every three consecutive elements having its maximum within 3 of its minimum.

Original entry on oeis.org

1, 1, 2, 6, 24, 36, 36, 48, 80, 128, 200, 288, 432, 648, 992, 1496, 2272, 3424, 5192, 7840, 11880, 17952, 27184, 41096, 62208, 94072, 142368, 215328, 325832, 492864, 745736, 1128096, 1706800, 2582024, 3906464, 5909784, 8941024, 13526368, 20464072, 30959200

Offset: 0

R. H. Hardin, Oct 23 2015

Alois P. Heinz, Table of n, a(n) for n = 0..5558

Diagonal of A263693.

Cf. A333833.

Empirical: a(n) = a(n-1) +a(n-2) -a(n-3) +a(n-4) for n>12.

G.f.: (8*x^12+8*x^11-4*x^10-8*x^8+6*x^7+20*x^6-7*x^5-16*x^4-4*x^3-1) / (x^4-x^3+x^2+x-1). - Alois P. Heinz, Apr 08 2020

a(0), a(24)-a(39) from Alois P. Heinz, Apr 08 2020

A263696 Number of length n arrays of permutations of 0..n-1 with each element moved by -n to n places and every three consecutive elements having its maximum within 4 of its minimum.

Original entry on oeis.org

1, 1, 2, 6, 24, 120, 288, 456, 768, 1508, 3434, 7560, 16024, 31872, 61704, 122464, 246296, 500506, 1015010, 2041372, 4098824, 8224718, 16535378, 33298104, 67036986, 134932118, 271438490, 545959366, 1098340012, 2209778140

Offset: 0

R. H. Hardin, Oct 24 2015

nonn

			Some solutions for n=7
..3....4....4....6....0....0....4....0....4....0....0....6....1....2....1....1
..0....6....0....5....3....1....3....2....0....1....4....5....0....6....0....3
..2....2....1....4....1....2....6....1....3....3....1....4....3....5....3....0
..1....5....2....2....4....4....5....4....1....2....3....2....4....4....4....2
..4....1....5....0....2....6....2....5....5....5....5....0....5....3....6....4
..5....3....6....3....5....3....1....3....2....6....2....1....2....1....2....6
..6....0....3....1....6....5....0....6....6....4....6....3....6....0....5....5

Diagonal of A263703.

Cf. A333833.

a(0), a(23)-a(29) from Alois P. Heinz, Apr 08 2020

A338614 Number of permutations p of [n] such that |p(i) - p(i-1)| <= 3 and |p(i) - p(i-2)| <= 4.

Original entry on oeis.org

1, 1, 2, 6, 24, 72, 124, 210, 394, 810, 1652, 3168, 5816, 10640, 19794, 37292, 70298, 131618, 245146, 456430, 851670, 1592008, 2976326, 5559808, 10379010, 19374184, 36175422, 67562524, 126185322, 235650426, 440038528

Offset: 0

Alois P. Heinz, Nov 03 2020

Wikipedia, Permutation

Cf. A263690, A263696, A263745, A333833, A338738.

b:= proc(s, x, y) option remember; `if`(s={}, 1, add(
      `if`((x=0 or abs(x-j)<=4) and (y=0 or abs(y-j)<=3),
          b(s minus {j}, y, j), 0), j=s))
    end:
a:= n-> b({$1..n}, 0$2):
seq(a(n), n=0..20);

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A363181 Number of permutations p of [n] such that for each i in [n] we have: (i>1) and |p(i)-p(i-1)| = 1 or (i

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A263690 Number of length n arrays of permutations of 0..n-1 with each element moved by -n to n places and every three consecutive elements having its maximum within 3 of its minimum.

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A263696 Number of length n arrays of permutations of 0..n-1 with each element moved by -n to n places and every three consecutive elements having its maximum within 4 of its minimum.

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A338614 Number of permutations p of [n] such that |p(i) - p(i-1)| <= 3 and |p(i) - p(i-2)| <= 4.

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