A075852 -id:A075852 - cp's OEIS frontend

A001883 Number of permutations s of {1,2,...,n} such that |s(i)-i|>1 for each i=1,2,...,n.

1, 0, 0, 0, 1, 4, 29, 206, 1708, 15702, 159737, 1780696, 21599745, 283294740, 3995630216, 60312696452, 970234088153, 16571597074140, 299518677455165, 5711583170669554, 114601867572247060, 2413623459384988298, 53238503492701261201, 1227382998752177970288, 29520591675204638641249

Offset: 0

N. J. A. Sloane

nonn

Permanent of the (0,1)-matrix having (i,j)-th entry equal to 0 iff this is on the first lower-diagonal, diagonal or the first upper-diagonal. - Simone Severini, Oct 14 2004

J. Riordan, "The enumeration of permutations with three-ply staircase restrictions," unpublished memorandum, Bell Telephone Laboratories, Murray Hill, NJ, Oct 1963.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Eric M. Schmidt, Table of n, a(n) for n = 0..200
Dmitry Efimov, Determinants of generalized binary band matrices, arXiv:1702.05655 [math.RA], 2017.
J. Riordan, Letter, Oct 31 1977
N. J. A. Sloane, Annotated copy of Riordan's Three-Ply Staircase paper
Donovan Young, The Number of Domino Matchings in the Game of Memory, J. Int. Seq., Vol. 21 (2018), Article 18.8.1.
Donovan Young, Generating Functions for Domino Matchings in the 2 X k Game of Memory, J. Int. Seq., Vol. 22 (2019), Article 19.8.7.
D. Zeilberger, Automatic Enumeration of Generalized Menage Numbers, arXiv preprint arXiv:1401.1089 [math.CO], 2014.

Cf. A001884-A001891, A075851, A075852.
Also a diagonal of A080018.
Column k=0 of A323671.

b:= proc(n, s) option remember; `if`(n=0, 1, add(
      `if`(abs(n-i)<=1, 0, b(n-1, s minus {i})), i=s))
    end:
a:= n-> b(n, {$1..n}):
seq(a(n), n=0..15);  # Alois P. Heinz, Jul 04 2015

b[n_, s_List] := b[n, s] = If[n == 0, 1, Sum[If[Abs[n-i] <= 1, 0, b[n-1, s ~Complement~ {i}]], {i, s}]]; a[n_] := b[n, Range[n]]; Table[Print[a[n]]; a[n], {n, 4, 24}] (* Jean-François Alcover, Nov 10 2015, after Alois P. Heinz *)

permRWNb(a)=n=matsize(a)[1]; if(n==1,return(a[1,1])); sg=1; in=vectorv(n); x=in; x=a[,n]-sum(j=1,n,a[,j])/2; p=prod(i=1,n,x[i]); for(k=1,2^(n-1)-1,sg=-sg; j=valuation(k,2)+1; z=1-2*in[j]; in[j]+=z; x+=z*a[,j]; p+=prod(i=1,n,x[i],sg)); return(2*(2*(n%2)-1)*p)
for(n=1,23,a=matrix(n,n,i,j,abs(i-j)>1);print1(permRWNb(a)",")) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), May 16 2007

a(n) = (n+1)*a(n-1) - (n-3)*a(n-2) - (n-4)*a(n-3) + (n-4)*a(n-4) + a(n-5) + (-1)^n * Lucas(n-3), n > 4. [Riordan] (Note: There is a slight mistake in Riordan's paper. On p. 3 it should say that a_5 = 3.) - Eric M. Schmidt, Oct 09 2017
From Vaclav Kotesovec, Oct 10 2017: (Start)
a(n) = n*a(n-1) + 4*a(n-2) - 3*(n-3)*a(n-3) + (n-4)*a(n-4) + 2*(n-5)*a(n-5) - (n-7)*a(n-6) - a(n-7).
a(n) ~ exp(-3) * n!.
(End)

More terms and better description from Reiner Martin, Oct 14 2002
More terms from Vladimir Baltic, Vladeta Jovovic, Jan 04 2003
More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), May 16 2007
a(22)-a(24) from Alois P. Heinz, Jul 04 2015
a(0)-a(3) from Eric M. Schmidt, Oct 09 2017

A075851 Number of permutations s of {1,2,...,n} such that |s(i)-i|>2 for each i=1,2,...,n.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 8, 112, 1168, 13365, 159414, 2036488, 27780408, 404351752, 6263006598, 102946702825, 1790795492176, 32880327473840, 635630231970048, 12907624693811937, 274744151265431700, 6117666413618771968, 142238172767973342656

Offset: 0

Reiner Martin, Oct 15 2002

nonn

a(n) equals the permanent of the n X n matrix with 0's along the main diagonal, the superdiagonal, the subdiagonal, the sub-subdiagonal, the super-superdiagonal, and 1's everywhere else. - John M. Campbell, Jul 09 2011

Alois P. Heinz, Table of n, a(n) for n = 0..450

Cf. A001883, A075852.

b:= proc(s) option remember; (n-> `if`(n=0, 1, add(
      `if`(abs(n-i)>2, b(s minus {i}), 0), i=s)))(nops(s))
    end:
a:= n-> b({$1..n}):
seq(a(n), n=0..15);  # Alois P. Heinz, Jan 25 2019

a[0] = 1; a[n_] := a[n] = If[n<6, 0, SparseArray[{Band[{1, 1}] -> 0, Band[{2, 1}] -> 0, Band[{3, 1}] -> 0, Band[{1, 2}] -> 0, Band[{1, 3}] -> 0}, {n, n}, 1] // Permanent];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 23}] (* Jean-François Alcover, Apr 30 2019 *)

More terms from Vladimir Baltic, Vladeta Jovovic, Jan 04 2003
a(21) from Alois P. Heinz, Jul 04 2015
a(22)-a(23) from Alois P. Heinz, Jan 22 2019
a(0)=1 prepended by Alois P. Heinz, Jan 25 2019

A183244 T(n,k) = Number of permutations of 1..n+2*k-1 with each element displaced by at least k.

Original entry on oeis.org

1, 1, 2, 1, 4, 9, 1, 8, 29, 44, 1, 16, 112, 206, 265, 1, 32, 436, 1168, 1708, 1854, 1, 64, 1708, 6984, 13365, 15702, 14833, 1, 128, 6724, 41808, 114124, 159414, 159737, 133496, 1, 256, 26572, 250464, 998112, 1799688, 2036488, 1780696, 1334961, 1, 512

Offset: 1

R. H. Hardin, Jan 03 2011

Table starts
........1.........1..........1............1.............1...............1
........2.........4..........8...........16............32..............64
........9........29........112..........436..........1708............6724
.......44.......206.......1168.........6984.........41808..........250464
......265......1708......13365.......114124........998112.........8751552
.....1854.....15702.....159414......1799688......21201024.......252813312
....14833....159737....2036488.....29125117.....441629332......6860776320
...133496...1780696...27780408....486980182....9154333160....178195229760
..1334961..21599745..404351752...8490078104..192565379941...4564491262444
.14684570.283294740.6263006598.154750897552.4146526612518.116967725946488

			All permutations of 1-5 with minimum displacement 2:
(3,4,5,1,2) (3,4,5,2,1) (4,5,1,2,3) (5,4,1,2,3).

R. H. Hardin, Table of n, a(n) for n = 1..201

Column 1 is A000166(n+1).
Column 2 is A001883(n+3).
Column 3 is A075851(n+5).
Column 4 is A075852(n+7).

T[n_, k_] := Permanent[nrows = n+2k-1; Table[If[Abs[i-j] <= k-1, 0, 1], {i, 1, nrows}, {j, 1, nrows}]]; Table[t = T[n-k+1, k]; Print[ "T(", n-k+1, ",", k, ") = ", t]; t, {n, 1, 9}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Jan 07 2016, adapted from Sage *)

def A183244_T(n,k):
    return Matrix(lambda i,j: 0 if abs(i-j) <= (k-1) else 1, nrows=n+2*k-1).permanent() # D. S. McNeil, Jan 04 2011

A299789 Number T(n,k) of permutations p of [n] such that min_{j=1..n} |p(j)-j| = k; triangle T(n,k), n >= 0, 0 <= k <= floor(n/2), read by rows.

Original entry on oeis.org

0, 1, 1, 1, 4, 2, 15, 8, 1, 76, 40, 4, 455, 236, 28, 1, 3186, 1648, 198, 8, 25487, 13125, 1596, 111, 1, 229384, 117794, 14534, 1152, 16, 2293839, 1175224, 146372, 12929, 435, 1, 25232230, 12903874, 1621282, 152430, 6952, 32, 302786759, 154615096, 19563257, 1922364, 112416, 1707, 1

Offset: 0

Alois P. Heinz, Jan 21 2019

			T(4,0) = 15: 1234, 1243, 1324, 1342, 1423, 1432, 2134, 2314, 2431, 3124, 3214, 3241, 4132, 4213, 4231.
T(4,1) = 8: 2143, 2341, 2413, 3142, 3421, 4123, 4312, 4321.
T(4,2) = 1: 3412.
T(5,2) = 4: 34512, 34521, 45123, 54123.
T(6,3) = 1: 456123.
T(7,3) = 8: 4567123, 4567132, 4567213, 4567231, 5671234, 5761234, 6571234, 7561234.
T(8,4) = 1: 56781234.
T(9,4) = 16: 567891234, 567891243, 567891324, 567891342, 567892134, 567892143, 567892314, 567892341, 678912345, 679812345, 687912345, 697812345, 768912345, 769812345, 867912345, 967812345.
Triangle T(n,k) begins:
          0;
          1;
          1,         1;
          4,         2;
         15,         8,        1;
         76,        40,        4;
        455,       236,       28,       1;
       3186,      1648,      198,       8;
      25487,     13125,     1596,     111,      1;
     229384,    117794,    14534,    1152,     16;
    2293839,   1175224,   146372,   12929,    435,    1;
   25232230,  12903874,  1621282,  152430,   6952,   32;
  302786759, 154615096, 19563257, 1922364, 112416, 1707, 1;
  ...

Alois P. Heinz, Rows n = 0..21, flattened

Columns k=0-1 give: A002467, A296050.
Row sums give A000142 (for n>0).
T(2n,n) gives A057427.
T(2n+1,n) gives A000079.
T(2n+2,n) gives A306545.
Cf. A000166, A001883, A075851, A075852, A129118, A130152, A306543.

b:= proc(s) option remember; (n-> `if`(n=1, x^(s[1]-1),
      add((p-> add(coeff(p, x, i)*x^min(i, abs(n-j)),
      i=0..degree(p)))(b(s minus {j})), j=s)))(nops(s))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..max(0, degree(p))))(b({$1..n})):
seq(T(n), n=0..14);
# second Maple program:
A:= proc(n, k) option remember; `if`(n=0, 0, LinearAlgebra[
      Permanent](Matrix(n, (i, j)-> `if`(abs(i-j)>=k, 1, 0))))
    end:
T:= (n, k)-> A(n, k)-A(n, k+1):
seq(seq(T(n, k), k=0..n/2), n=0..14);

A[n_, k_] := A[n, k] = If[n==0, 0, Permanent[Table[If[Abs[i-j] >= k, 1, 0], {i, 1, n}, {j, 1, n}]]];
T[n_, k_] := A[n, k] - A[n, k+1];
Table[T[n, k], {n, 0, 14}, {k, 0, n/2}] // Flatten (* Jean-François Alcover, May 01 2019, from 2nd Maple program *)

T(n,k) = A306543(n,k) - A306543(n,k+1) for n > 0.
Sum_{k=1..floor(n/2)} k * T(n,k) = A129118(n).
Sum_{k=1..floor(n/2)} T(n,k) = A000166(n).
Sum_{k=2..floor(n/2)} T(n,k) = A001883(n).
Sum_{k=3..floor(n/2)} T(n,k) = A075851(n).
Sum_{k=4..floor(n/2)} T(n,k) = A075852(n).

A001887 Number of permutations p of {1,2,...,n} such that p(i) - i < 0 or p(i) - i > 2 for all i.

Original entry on oeis.org

1, 0, 0, 0, 1, 5, 33, 236, 1918, 17440, 175649, 1942171, 23396353, 305055960, 4280721564, 64330087888, 1030831875953, 17545848553729, 316150872317105, 6012076099604308, 120330082937778554

Offset: 0

N. J. A. Sloane

nonn

Previous name was: Hit polynomials.

J. Riordan, The enumeration of permutations with three-ply staircase restrictions, unpublished memorandum, Bell Telephone Laboratories, Murray Hill, NJ, Oct 1963. (See A001883)
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..300
E. R. Canfield, N. C. Wormald, Menage numbers, bijections and P-recursiveness, Discr. Math. 63 (1987) 117, table Section 7.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 373
V. Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 224.
N. J. A. Sloane, Annotated copy of Riordan's Three-Ply Staircase paper (unpublished memorandum, Bell Telephone Laboratories, Murray Hill, NJ, Oct 1963)
D. Zeilberger, Automatic Enumeration of Generalized Menage Numbers, arXiv preprint arXiv:1401.1089 [math.CO], 2014.

Cf. A000255, A000271, A001883-A001891, A075851, A075852.

First column of A080061.

nmax = 21;
gf = 1/(x^2-1)(x-Sum[n! (x(x-1)/(x^3-2x-1))^n + O[x]^nmax, {n, 0, nmax}]);
CoefficientList[gf, x] (* Jean-François Alcover, Aug 19 2018 *)

G.f.: (1/(x^2-1))*(x-Sum_{n>=0} n!*(x*(x-1)/(x^3-2*x-1))^n). - Vladeta Jovovic, Jun 30 2007
D-finite with recurrence (P. Flajolet, 1997): a(n) = (n-1)*a(n-1) + (n+2)*a(n-2) - (3*n-13)*a(n-3) - (2*n-8)*a(n-4) + (3*n-15)*a(n-5) + (n-4)*a(n-6) - (n-7)*a(n-7) - a(n-8), n>8.
a(n) ~ exp(-3) * n!. - Vaclav Kotesovec, Sep 10 2014

More terms from Vladimir Baltic and Vladeta Jovovic, Jan 05 2003

New name from Vaclav Kotesovec using a former comment by Vladimir Baltic and Vladeta Jovovic, Sep 16 2014

A078509 Number of permutations p of {1,2,...,n} such that p(i)-i != 1 and p(i)-i != 2 for all i.

Original entry on oeis.org

1, 1, 1, 1, 5, 23, 131, 883, 6859, 60301, 591605, 6405317, 75843233, 974763571, 13512607303, 200949508327, 3190881283415, 53880906258521, 964039575154409, 18217997734199113, 362584510633666621, 7580578211464070863, 166099466140519353035, 3806162403831340850651

Offset: 0

Vladimir Baltic, Vladeta Jovovic, Jan 05 2003

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..450

Cf. A000255, A055790, A001883, A001887, A075851, A075852.

a:= proc(n) option remember; `if`(n<4, 1,
      (n-1)*a(n-1) +(n-3)*a(n-2) +a(n-3))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Jan 10 2014

a = DifferenceRoot[Function[{y, n}, {-y[n] - n y[n+1] - (n+2) y[n+2] + y[n+3] == 0, y[0] == 1, y[1] == 1, y[2] == 1, y[3] == 1}]];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Dec 20 2020, after Alois P. Heinz *)

From Vladeta Jovovic, Jul 16 2007: (Start)
G.f.: x/(1+x)*Sum_{n>=0} (n+1)!*(x/(1+x)^2)^n.
a(n) = Sum_{k=1..n} (-1)^(n-k)*k!*binomial(n+k-2,2*k-2). (End)
a(n) ~ exp(-2) * n!. - Vaclav Kotesovec, Aug 25 2014

More terms from Alois P. Heinz, Jan 10 2014

A306543 Number T(n,k) of permutations p of [n] such that |p(j)-j| >= k (for all j in [n]); triangle T(n,k), n >= 0, 0 <= k <= floor(n/2), read by rows.

Original entry on oeis.org

1, 1, 2, 1, 6, 2, 24, 9, 1, 120, 44, 4, 720, 265, 29, 1, 5040, 1854, 206, 8, 40320, 14833, 1708, 112, 1, 362880, 133496, 15702, 1168, 16, 3628800, 1334961, 159737, 13365, 436, 1, 39916800, 14684570, 1780696, 159414, 6984, 32, 479001600, 176214841, 21599745, 2036488, 114124, 1708, 1

Offset: 0

Alois P. Heinz, Feb 22 2019

			Triangle T(n,k) begins:
          1;
          1;
          2,         1;
          6,         2;
         24,         9,        1;
        120,        44,        4;
        720,       265,       29,       1;
       5040,      1854,      206,       8;
      40320,     14833,     1708,     112,      1;
     362880,    133496,    15702,    1168,     16;
    3628800,   1334961,   159737,   13365,    436,    1;
   39916800,  14684570,  1780696,  159414,   6984,   32;
  479001600, 176214841, 21599745, 2036488, 114124, 1708, 1;
  ...

Alois P. Heinz, Rows n = 0..22, flattened
Wikipedia, Permutation

Columns k=0-6 give (offsets may differ): A000142, A000166, A001883, A075851, A075852, A183242, A183243.
T(2n,n) gives A000012.
T(2n+1,n) gives A000079.
T(2n+2,n) gives A183245 for n > 0.
T(2n+3,n) gives A183246 for n > 0.
T(2n+4,n) gives A183247 for n > 0.
Cf. A183244, A299789.

T:= proc(n, k) option remember; `if`(n=0, 1, LinearAlgebra[
      Permanent](Matrix(n, (i, j)-> `if`(abs(i-j)>=k, 1, 0))))
    end:
seq(seq(T(n, k), k=0..floor(n/2)), n=0..12);

T[n_, k_] := T[n, k] = If[n==0, 1, Permanent[Table[
     If[Abs[i-j] >= k, 1, 0], {i, n}, {j, n}]]];
Table[Table[T[n, k], {k, 0, Floor[n/2]}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Mar 26 2021, after Alois P. Heinz *)

T(n,k) = Sum_{j=k..floor(n/2)} A299789(n,j) for n > 0.

cp's OEIS Frontend

A001883 Number of permutations s of {1,2,...,n} such that |s(i)-i|>1 for each i=1,2,...,n.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A075851 Number of permutations s of {1,2,...,n} such that |s(i)-i|>2 for each i=1,2,...,n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

A183244 T(n,k) = Number of permutations of 1..n+2*k-1 with each element displaced by at least k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Sage

A299789 Number T(n,k) of permutations p of [n] such that min_{j=1..n} |p(j)-j| = k; triangle T(n,k), n >= 0, 0 <= k <= floor(n/2), read by rows.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A001887 Number of permutations p of {1,2,...,n} such that p(i) - i < 0 or p(i) - i > 2 for all i.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A078509 Number of permutations p of {1,2,...,n} such that p(i)-i != 1 and p(i)-i != 2 for all i.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A306543 Number T(n,k) of permutations p of [n] such that |p(j)-j| >= k (for all j in [n]); triangle T(n,k), n >= 0, 0 <= k <= floor(n/2), read by rows.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple