A008305 -id:A008305 - cp's OEIS frontend

A004306 Rook polynomials.

1, 1, 2, 6, 24, 44, 80, 144, 264, 484, 888, 1632, 3000, 5516, 10144, 18656, 34312, 63108, 116072, 213488, 392664, 722220, 1328368, 2443248, 4493832, 8265444, 15202520, 27961792, 51429752, 94594060, 173985600, 320009408, 588589064, 1082584068, 1991182536

Offset: 0

N. J. A. Sloane

a(n) is the number of perfect matchings in the circulant graph with 2*n vertices with jumps 1 and 3. - Robert Israel, Jan 24 2019

D. H. Lehmer, Permutations with strongly restricted displacements. Combinatorial theory and its applications, II (Proc. Colloq., Balatonfured, 1969), pp. 755-770. North-Holland, Amsterdam, 1970.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..400
N. Metropolis, M. L. Stein, and P. R. Stein, Permanents of cyclic (0,1) matrices, Journal of Combinatorial Theory, Volume 7, Issue 4, December 1969, Pages 291-321.
Earl Glen Whitehead, Jr., Four-discordant permutations, J. Austral. Math. Soc. Ser. A 28 (1979), no. 3, 369-377.
Index entries for linear recurrences with constant coefficients, signature (2,0,0,-1).

Cf. A000803. 4th column of A008305.

Equals 2 * (A001644(n) + 1), n>3.

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1-x+ 2*x^3+13*x^4-3*x^5-6*x^6-10*x^7)/(1-2*x+x^4) )); // G. C. Greubel, Apr 22 2019

Join[{1,1,2,6},LinearRecurrence[{2,0,0,-1},{24,44,80,144},40]] (* or *) CoefficientList[ Series[ (1-x+2x^3+13x^4- 3x^5- 6x^6- 10x^7)/ (1-2x+ x^4),{x,0,40}],x] (* Harvey P. Dale, Dec 13 2011 *)

my(x='x+O('x^40)); Vec((1-x+2*x^3+13*x^4-3*x^5-6*x^6-10*x^7)/(1 -2*x+x^4)) \\ G. C. Greubel, Apr 22 2019

((1-x+2*x^3+13*x^4-3*x^5-6*x^6-10*x^7)/(1-2*x+x^4)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Apr 22 2019

G.f.: (1 - x + 2*x^3 + 13*x^4 - 3*x^5 - 6*x^6 - 10*x^7)/(1 - 2*x + x^4).

a(n) = 2*a(n-1) - a(n-4); a(0)=1, a(1)=1, a(2)=2, a(3)=6, a(4)=24, a(5)=44, a(6)=80, a(7)=144. - Harvey P. Dale, Dec 13 2011

A000382 Restricted permutations.

Original entry on oeis.org

6, 11, 20, 36, 65, 119, 218, 400, 735, 1351, 2484, 4568, 8401, 15451, 28418, 52268, 96135, 176819, 325220, 598172, 1100209, 2023599, 3721978, 6845784, 12591359, 23159119, 42596260, 78346736, 144102113, 265045107, 487493954

Offset: 4

N. J. A. Sloane

The fourth column of A008305, divided by 4.

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).

Vincenzo Librandi, Table of n, a(n) for n = 4..1000
N. S. Mendelsohn, Permutations with confined displacement, Canad. Math. Bull., 4 (1961), 29-38.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992

Cf. A008305, A000496 divided by 4, A020992.

A000382:=-(-6+z+2*z**2+4*z**3+z**4)/(z-1)/(z**3+z**2+z-1); [Conjectured by Simon Plouffe in his 1992 dissertation.]
a:= n-> if n<4 then 0 elif n=4 then 6 else (Matrix([[11,7,4,2]]). Matrix(4, (i,j)-> if (i=j-1) then 1 elif j=1 then [2,0,0,-1][i] else 0 fi)^(n-2))[1,4] fi: seq(a(n), n=4..30); # Alois P. Heinz, Aug 26 2008

a[n_] := Which[n<4, 0, n == 4, 6, True, {11, 7, 4, 2}.MatrixPower[Table[Which[i == j-1, 1, j == 1, {2, 0, 0, -1}[[i]], True, 0], {i, 1, 4}, {j, 1, 4}], n-2] // Last]; Table[a[n], {n, 4, 27}] (* Jean-François Alcover, Mar 12 2014, after Alois P. Heinz *)

a(n) = a(n-1)+a(n-2)+a(n-3)-2 (conjectured).

More terms from Vincenzo Librandi, Mar 14 2014

A321352 Triangle T(n,k) giving the number of permutations pi of {1,2,...,n} such that for all i, pi(i) is not in {i, i+1, ..., i+k-1} (mod n), with 0 <= k <= n - 1.

Original entry on oeis.org

1, 2, 1, 6, 2, 1, 24, 9, 2, 1, 120, 44, 13, 2, 1, 720, 265, 80, 20, 2, 1, 5040, 1854, 579, 144, 31, 2, 1, 40320, 14833, 4738, 1265, 264, 49, 2, 1, 362880, 133496, 43387, 12072, 2783, 484, 78, 2, 1, 3628800, 1334961, 439792, 126565, 30818, 6208, 888, 125, 2, 1

Offset: 1

Peter Kagey, Feb 25 2020

This is A008305 with the rows reversed.
First column is A000142 (factorial numbers).
Second column is A000166 (derangements).
Third column is A000179 (ménage numbers).
Fourth column is A000183 (discordant permutations)

			Table begins:
       1
       2,      1
       6,      2,     1
      24,      9,     2,     1
     120,     44,    13,     2,    1
     720,    265,    80,    20,    2,   1
    5040,   1854,   579,   144,   31,   2,  1
   40320,  14833,  4738,  1265,  264,  49,  2, 1
  362880, 133496, 43387, 12072, 2783, 484, 78, 2, 1

Peter Kagey, Table of n, a(n) for n = 1..276 (first 23 rows, flattened)

Cf. A008305.

Cf. A000142, A000166, A000179, A000183, A004307, A189389, A184965.

A048162 Expansion of (1 - x + 3x^3 - 2x^4 - 3x^5)/(1 - 2x + x^3).

Original entry on oeis.org

1, 1, 2, 6, 9, 13, 20, 31, 49, 78, 125, 201, 324, 523, 845, 1366, 2209, 3573, 5780, 9351, 15129, 24478, 39605, 64081, 103684, 167763, 271445, 439206, 710649, 1149853, 1860500, 3010351, 4870849, 7881198, 12752045, 20633241, 33385284

Offset: 0

N. J. A. Sloane

nonn

Number of permutations of 1..n such that each position is fixed or moves to an adjacent position (with n considered adjacent to 1). For example, a(4) = 9 because there is the identity; 2 cyclic permutations; 4 swaps of one pair of adjacent entries; and 2 swaps of two pairs of adjacent entries. - Joshua Zucker, Nov 13 2003

Lehmer, D. H.; Permutations with strongly restricted displacements. Combinatorial theory and its applications, II (Proc. Colloq., Balatonfured, 1969), pp. 755-770. North-Holland, Amsterdam, 1970.

Index entries for linear recurrences with constant coefficients, signature (2, 0, -1).

3rd column of A008305.

Cf. A001610.

CoefficientList[Series[(1-x+3x^3-2x^4-3x^5)/(1-2x+x^3),{x,0,40}],x] (* or *) Join[{1,1,2},#[[3]]+#[[1]]+2&/@Partition[Fibonacci[Range[2,50]],3,1]] (* Harvey P. Dale, Apr 06 2017 *)

For n>4, a(n) = a(n-1) + a(n-2) - 2. - Joshua Zucker, Nov 13 2003
a(n) = Fibonacci(n+1) + Fibonacci(n-1) + 2, for n>2. - Jessa Lee (jessal(AT)comcast.net), Nov 25 2003
For n > 2, a(n)=A001610(n-1) - 3. - Toby Gottfried, Apr 13 2013

Second formula corrected by David Radcliffe, Jan 16 2011

A345461 Triangle T(n,k) (n >= 1, 0 <= k <= n-1) read by rows: number of distinct permutations after k steps of the "optimist" algorithm.

Original entry on oeis.org

1, 2, 1, 6, 1, 1, 24, 6, 1, 1, 120, 38, 7, 1, 1, 720, 232, 53, 7, 1, 1, 5040, 1607, 404, 74, 7, 1, 1, 40320, 12984, 3383, 732, 108, 7, 1, 1, 362880, 117513, 31572, 7043, 1292, 167, 9, 1, 1, 3628800, 1182540, 324112, 75350, 14522, 2384, 260, 11, 1, 1

Offset: 1

Olivier Gérard, Jun 20 2021

Start with the n! permutations of order n. Apply an iteration of the "optimist" sorting algorithm. Count the distinct permutations, until all are sorted.
The length of each row is n.
The optimist algorithm is: rotate right all currently unsorted letters by the distance between the first unsorted one and its sorted position. An example is given in A345453.

			Triangle begins:
.
     1;
     2,     1;
     6,     1,    1;
    24,     6,    1,   1;
   120,    38,    7,   1,  1;
   720,   232,   53,   7,  1,  1;
  5040,  1607,  404,  74,  7,  1,  1;
.

Cf. A345453 (permutations according to number of steps for sorting).
Cf. A321352 and A008305 (the equivalent for Eulerian numbers).
Cf. A345462 (the equivalent for Stirling numbers of 1st kind).
Cf. A345464 (first column).

T(n,0) = n!; T(n,n-1) = 1; T(n,n-2) = 1 for n > 2.

A306738 Number of permutations of [n] allowing i->i+j (mod n), j=0..ceiling(n/2)-1.

Original entry on oeis.org

1, 1, 1, 2, 2, 13, 20, 144, 264, 2783, 6208, 79118, 204448, 3182225, 9411840, 170576156, 566725760, 11804363619, 43573050368, 1023207993178, 4152609019392, 108681408827381, 481065936784384, 13880706183899752

Offset: 0

Alois P. Heinz, Mar 06 2019

nonn

Cf. A008305.

a(n) = A008305(n,ceiling(n/2)).

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A004306 Rook polynomials.

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A000382 Restricted permutations.

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A321352 Triangle T(n,k) giving the number of permutations pi of {1,2,...,n} such that for all i, pi(i) is not in {i, i+1, ..., i+k-1} (mod n), with 0 <= k <= n - 1.

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A048162 Expansion of (1 - x + 3*x^3 - 2*x^4 - 3*x^5)/(1 - 2*x + x^3).

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A345461 Triangle T(n,k) (n >= 1, 0 <= k <= n-1) read by rows: number of distinct permutations after k steps of the "optimist" algorithm.

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Author

Keywords

Comments

Examples

Crossrefs

Formula

A306738 Number of permutations of [n] allowing i->i+j (mod n), j=0..ceiling(n/2)-1.

Views

Author

Keywords

Crossrefs

Formula

A048162 Expansion of (1 - x + 3x^3 - 2x^4 - 3x^5)/(1 - 2x + x^3).