A165960 -id:A165960 - cp's OEIS frontend

A165961 Number of circular permutations of length n without 3-sequences.

1, 5, 20, 102, 627, 4461, 36155, 328849, 3317272, 36757822, 443846693, 5800991345, 81593004021, 1228906816941, 19733699436636, 336554404751966, 6075478765948135, 115734570482611885, 2320148441078578447, 48827637296350480457, 1076313671861962141616

Offset: 3

Isaac Lambert, Oct 01 2009

nonn

Circular permutations are permutations whose indices are from the ring of integers modulo n. 3-sequences are of the form i,i+1,i+2. Sequence gives number of permutations of [n] starting with 1 and having no 3-sequences.

a(n) is also the number of permutations of length n-1 without consecutive fixed points (cf. A180187). - David Scambler, Mar 27 2011

			For n=4 the a(4)=5 solutions are (0,1,3,2), (0,2,1,3), (0,2,3,1), (0,3,1,2) and (0,3,2,1).

Wayne M. Dymacek, Isaac Lambert and Kyle Parsons, Arithmetic Progressions in Permutations, http://math.ku.edu/~ilambert/CN.pdf, 2012. - From N. J. A. Sloane, Sep 15 2012 [broken link]

Michael De Vlieger, Table of n, a(n) for n = 3..450
Wayne M. Dymacek and Isaac Lambert, Permutations Avoiding Runs of i, i+1, i+2 or i, i-1, i-2, Journal of Integer Sequences, Vol. 14 (2011), Article 11.1.6.
Kyle Parsons, Arithmetic progressions in permutations, Thesis, 2011.

Cf. A002628, A165960, A165962.
Cf. A000166, A180186, - Emeric Deutsch, Sep 07 2010
A column of A216718. - N. J. A. Sloane, Sep 15 2012

d[0] := 1: for n to 51 do d[n] := n*d[n-1]+(-1)^n end do: a := proc (n) options operator, arrow: sum(binomial(n-k, k)*d[n-k-1], k = 0 .. floor((1/2)*n)) end proc: seq(a(n), n = 3 .. 23); # Emeric Deutsch, Sep 07 2010

a[n_] := Sum[Binomial[n-k, k] Subfactorial[n-k-1], {k, 0, n/2}];
a /@ Range[3, 21] (* Jean-François Alcover, Oct 29 2019 *)

Let b(n) be the sequence A002628. Then for n > 5, this sequence satisfies a(n) = b(n-1) - b(n-3) + a(n-3).

a(n) = Sum_{k=0..n/2} binomial(n-k,k)*d(n-k-1), where d(j)=A000166(j) are the derangement numbers. - Emeric Deutsch, Sep 07 2010

More terms from Emeric Deutsch, Sep 07 2010

Edited by N. J. A. Sloane, Apr 04 2011

A165962 Number of circular permutations of length n without modular 3-sequences.

Original entry on oeis.org

1, 5, 18, 95, 600, 4307, 35168, 321609, 3257109, 36199762, 438126986, 5736774126, 80808984725, 1218563180295, 19587031966352, 334329804347219, 6039535339644630, 115118210694558105, 2308967760171049528, 48613722701436777455, 1072008447320752890459

Offset: 3

Isaac Lambert, Oct 01 2009

nonn

Circular permutations are permutations whose indices are from the ring of integers modulo n. Modular 3-sequences are of the following form: i,i+1,i+2, where arithmetic is modulo n.

			For n=4 the a(4)=5 solutions are (0,1,3,2), (0,2,1,3), (0,2,3,1), (0,3,1,2) and (0,3,2,1).

Wayne M. Dymacek, Isaac Lambert and Kyle Parsons, Arithmetic Progressions in Permutations, http://math.ku.edu/~ilambert/CN.pdf, 2012. - N. J. A. Sloane, Sep 15 2012

Max Alekseyev, Table of n, a(n) for n = 3..100

Cf. A002628, A165960, A165961.

First column of A216722. Cf. A216723. - N. J. A. Sloane, Sep 15 2012

f[i_,n_,k_]:=If[i==0&&k==0,1,If[i==n&&n==k,1,Binomial[k-1,k-i]*Binomial[n-k-1,k-i-1]+2*Binomial[k-1,k-i-1]*Binomial[n-k-1,k-i-1]+Binomial[k-1,k-i-1]*Binomial[n-k-1,k-i]]];
w1[i_,n_,k_]:=If[n-2k+i<0,0,If[n-2k+i==0,1,(n-2k+i-1)!]];
a[n_,k_]:=Sum[f[i,n,k]*w1[i,n,k],{i,0,k}];
A165962[n_]:=(n-1)!+Sum[(-1)^k*a[n,k],{k,1,n}];
Table[A165962[n],{n,3,23}] (* David Scambler, Sep 18 2012 *)

This sequence can be related to A165961 by the use of auxiliary sequences (and the auxiliary sequences can themselves be calculated by recurrence relations).

A002628 Number of permutations of length n without 3-sequences.

Original entry on oeis.org

1, 1, 2, 5, 21, 106, 643, 4547, 36696, 332769, 3349507, 37054436, 446867351, 5834728509, 82003113550, 1234297698757, 19809901558841, 337707109446702, 6094059760690035, 116052543892621951, 2325905946434516516, 48937614361477154273, 1078523843237914046247

Offset: 0

N. J. A. Sloane

nonn

a(n) = sum of row n of A180185. - Emeric Deutsch, Sep 06 2010

			a(4) = 21 because only 1234, 2341, and 4123 contain 3-sequences. - _Emeric Deutsch_, Sep 06 2010

Jackson, D. M.; Reilly, J. W. Permutations with a prescribed number of p-runs. Ars Combinatoria 1 (1976), number 1, 297-305.
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..450
D. M. Jackson and R. C. Read, A note on permutations without runs of given length, Aequationes Math. 17 (1978), no. 2-3, 336-343.
J. Riordan, Permutations without 3-sequences, Bull. Amer. Math. Soc., 51 (1945), 745-748.

Column k=0 of A047921.
Cf. A165960, A165961, A165962. - Isaac Lambert, Oct 07 2009
Cf. A000166, A180185. - Emeric Deutsch, Sep 06 2010

seq(coeff(convert(series(add(m!*((t-t^3)/(1-t^3))^m,m=0..50),t,50), polynom), t,n),n=0..25); # Pab Ter, Nov 06 2005
d[-1]:= 0: for n from 0 to 51 do d[n] := n*d[n-1]+(-1)^n end do: a:= proc(n) add(binomial(n-k, k)*(d[n-k]+d[n-k-1]), k = 0..floor((1/2)*n)) end proc: seq(a(n), n = 0..25); # Emeric Deutsch, Sep 06 2010
# third Maple program:
a:= proc(n) option remember; `if`(n<5,
      [1$2, 2, 5, 21][n+1], (n-3)*a(n-1)+(3*n-6)*a(n-2)+
      (4*n-12)*a(n-3)+(3*n-12)*a(n-4)+(n-5)*a(n-5))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Jul 21 2019

d[0] = 1; d[n_] := d[n] = n d[n - 1] + (-1)^n;
T[n_, k_] := If[n == 0 && k == 0, 1, If[k <= n/2, Binomial[n - k, k] d[n + 1 - k]/(n - k), 0]];
a[n_] := Sum[T[n, k], {k, 0, Quotient[n, 2]}];
a /@ Range[0, 25] (* Jean-François Alcover, May 23 2020 *)

a(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k)*(d(n-k) + d(n-k-1)) for n>0, where d(j) = A000166(j) are the derangement numbers. - Emeric Deutsch, Sep 06 2010

More terms from Pab Ter (pabrlos2(AT)yahoo.com), Nov 06 2005

a(0)=1 prepended by Alois P. Heinz, Jul 21 2019

A174073 Number of permutations of length n without modular consecutive triples i,i+2,i+4.

Original entry on oeis.org

1, 1, 2, 3, 24, 100, 594, 4389, 35744, 325395, 3288600, 36489992, 441091944, 5770007009, 81213883898, 1223895060315, 19662509172096, 335472890422812, 6057979283966814, 115434096553014565, 2314691409661484600, 48723117262650147387, 1074208020519754180896, 24755214452825129257168

Offset: 0

Isaac Lambert, Mar 06 2010

nonn

			For example, a(5) does not count the permutation (0,4,1,3,2) since 4,1,3 is an arithmetic progression of 2 mod(5).

Wayne M. Dymacek, Isaac Lambert and Kyle Parsons, Arithmetic Progressions in Permutations, http://math.ku.edu/~ilambert/CN.pdf, 2012.

Max Alekseyev, Table of n, a(n) for n = 0..100
Kyle Parsons, Arithmetic Progressions in Permutations, thesis, 2011. See Table 4 p. 12.

Column k=0 of A216724.

Cf. A165960, A174072, A174074, A174075.

a(11)-a(17) from Alois P. Heinz, Apr 13 2021

Terms a(18) onward from Max Alekseyev, Feb 04 2024

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A165961 Number of circular permutations of length n without 3-sequences.

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A165962 Number of circular permutations of length n without modular 3-sequences.

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A002628 Number of permutations of length n without 3-sequences.

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A174073 Number of permutations of length n without modular consecutive triples i,i+2,i+4.

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