A229730 -id:A229730 - 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

A229729 Number of separable permutations with exactly four occurrences of the 1-box pattern.

Original entry on oeis.org

0, 0, 0, 0, 8, 42, 178, 664, 2288

Offset: 0

N. J. A. Sloane, Oct 01 2013

See Kitaev-Remmel for precise definition.

S. Kitaev and J. Remmel, The 1-box pattern on pattern avoiding permutations, arXiv preprint arXiv:1305.6970 [math.CO], 2013.

Cf. A229730.

A346462 Triangle read by rows: T(n,k) gives the number of permutations of length n containing exactly k instances of the 1-box pattern; 0 <= k <= n.

Original entry on oeis.org

1, 1, 0, 0, 0, 2, 0, 0, 4, 2, 2, 0, 10, 4, 8, 14, 0, 40, 10, 42, 14, 90, 0, 230, 40, 226, 80, 54, 646, 0, 1580, 230, 1480, 442, 534, 128, 5242, 0, 12434, 1580, 11496, 2920, 4746, 1404, 498, 47622, 0, 110320, 12434, 101966, 22762, 45216, 13138, 7996, 1426

Offset: 0

Peter Kagey, Jul 19 2021

An instance of the 1-box pattern in a permutation pi is a letter pi_i such that pi_{i-1} or pi_{i+1} differs from pi_i by exactly 1.
Column k=0 is A002464. Columns k=2 and k=3 are given by A086852.
Main diagonal begins: 1,0,2,2,8,14,54,128,498,1426,5736,... A363181.

			The permutation 14327568 has 5 instances of the 1-box pattern:
- position 2 differs from position 3 by one,
- position 3 differs from positions 2 and 4 by one,
- position 4 differs from position 3 by one,
- position 6 differs from position 7 by one,
- position 7 differs from position 6 by one, and
positions 1, 5, and 8 differ from all of their neighbors by more than 1.
Table begins:
  n\k|  0  1    2   3    4   5   6
-----+-----------------------------
   0 |  1
   1 |  1  0
   2 |  0  0    2
   3 |  0  0    4   2
   4 |  2  0   10   4    8
   5 | 14  0   40  10   42  14
   6 | 90  0  230  40  226  80  54

Alois P. Heinz, Rows n = 0..20, flattened
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.

Cf. A002464, A086852, A229729, A229730, A363181.

Row sums give A000142.

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula

A229729 Number of separable permutations with exactly four occurrences of the 1-box pattern.

Views

Author

Keywords

Comments

Links

Crossrefs

A346462 Triangle read by rows: T(n,k) gives the number of permutations of length n containing exactly k instances of the 1-box pattern; 0 <= k <= n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs