A000440 -id:A000440 - cp's OEIS frontend

A000388 Number of permutations of an n-sequence discordant with three given permutations (see reference) in n-2 places.

6, 20, 180, 1106, 9292, 82980, 831545, 9139482, 109595496, 1423490744, 19911182207, 298408841160, 4770598226296, 81037124739588, 1457607971046492, 27675791180024802, 553166885187641670, 11609691036091870428, 255273744004170486155, 5868308906885934514178

Offset: 4

N. J. A. Sloane

nonn

J. Riordan, Discordant permutations, Scripta Math., 20 (1954), 14-23.
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).

J. Riordan, Discordant permutations, Scripta Math., 20 (1954), 14-23. [Annotated scanned copy]

Cf. A000500, A000470, A000440, A000476, A000380, A000492.

seq(f(n,2), n=5..30); # code for f(n,k) is given in A000440 - Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Feb 17 2001

sigma[t_, u_] = (1 - 2 t^2 (u^2) - 2 t^2 (1 + t) u^3 + 3 t^4 (u^4)) (1 - t* u)^(-1) (1 - (1 + 2 t) u - t *u^2 + t^3 (u^3))^(-1); ds[t_, n_] := D[sigma[t, u], {u, n}] /. u -> 0; su[n_] := su[n] = Sum[ Coefficient[ds[t, n]/n!, t, j]*(n - j)!*(y - 1)^j, {j, 0, n}]; f[n_, k_] := Coefficient[su[n], y, k]; Table[f[n, 2], {n, 4, 23}] (* Jean-François Alcover, Sep 01 2011, after Maple prog. *)

a(n) = coefficient of y^2 in sum_0^n sigma_{n, k}(n - k)!(y - 1)^k on y where the sigma_{n, k} have generating function sigma(t, u) = (1 - 2t^2(u^2) - 2t^2(1 + t)u^3 + 3t^4(u^4))(1 - tu)^(-1)(1 - (1 + 2t)u - tu^2 + t^3(u^3))^(-1). - Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Feb 17 2001

More terms from Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Feb 17 2001

A000380 Number of permutations of an n-sequence discordant with three given permutations (see reference) in n-3 places.

Original entry on oeis.org

6, 8, 40, 176, 1421, 10352, 93114, 912920, 9929997, 117970704, 1521176826, 21150414880, 315400444070, 5020920314016, 84979755347122, 1523710321272384, 28851091193764023, 575253584489378040, 12047084261153160394, 264377395040950523112, 6066972656940255290199

Offset: 3

N. J. A. Sloane

nonn

J. Riordan, Discordant permutations, Scripta Math., 20 (1954), 14-23.
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).

J. Riordan, Discordant permutations, Scripta Math., 20 (1954), 14-23. [Annotated scanned copy]

Cf. A000500, A000470, A000440, A000476, A000492, A000388.

seq(f(n,3), n=3..30); # code for f(n,k) is given in A000440 - Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Feb 17 2001

sigma[t_, u_] = (1 - 2 t^2 (u^2) - 2 t^2 (1 + t) u^3 + 3 t^4 (u^4)) (1 - t* u)^(-1) (1 - (1 + 2 t) u - t *u^2 + t^3 (u^3))^(-1); ds[t_, n_] := D[sigma[t, u], {u, n}] /. u -> 0; su[n_] := su[n] = Sum[ Coefficient[ds[t, n]/n!, t, j]*(n - j)!*(y - 1)^j, {j, 0, n}]; f[n_, k_] := Coefficient[su[n], y, k]; Table[f[n, 3], {n, 3, 23}] (* Jean-François Alcover, Sep 01 2011, after Maple prog. *)

a(n) = coefficient of y^3 in sum_0^n sigma_{n, k}(n - k)!(y - 1)^k on y where the sigma_{n, k} have generating function sigma(t, u) = (1 - 2t^2(u^2) - 2t^2(1 + t)u^3 + 3t^4(u^4))(1 - tu)^(-1)(1 - (1 + 2t)u - tu^2 + t^3(u^3))^(-1). - Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Feb 17 2001

More terms from Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Feb 17 2001

A000476 Number of permutations of an n-sequence discordant with three given permutations (see reference) in n-1 places.

Original entry on oeis.org

15, 72, 609, 4960, 46188, 471660, 5275941, 64146768, 842803767, 11902900380, 179857257960, 2895705788736, 49491631601635, 895010868095256, 17074867330880805, 342733960299356800, 7220616209235766260, 159312370008282356844, 3673720238903201471593

Offset: 5

N. J. A. Sloane

nonn

J. Riordan, Discordant permutations, Scripta Math., 20 (1954), 14-23.
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).

J. Riordan, Discordant permutations, Scripta Math., 20 (1954), 14-23. [Annotated scanned copy]

Cf. A000500, A000470, A000440, A000492, A000380, A000388.

seq(f(n,1), n=5..30); # where code for f(n,k) is given in A000440 - Barbara Haas Margolius (margolius(AT)math.csuohio.edu) Feb 17 2001

sigma[t_, u_] = (1 - 2 t^2 (u^2) - 2 t^2 (1 + t) u^3 + 3 t^4 (u^4)) (1 - t* u)^(-1) (1 - (1 + 2 t) u - t *u^2 + t^3 (u^3))^(-1); ds[t_, n_] := D[sigma[t, u], {u, n}] /. u -> 0; su[n_] := su[n] = Sum[ Coefficient[ds[t, n]/n!, t, j]*(n - j)!*(y - 1)^j, {j, 0, n}]; f[n_, k_] := Coefficient[su[n], y, k]; Table[f[n, 1], {n, 5, 23}] (* Jean-François Alcover, Sep 01 2011, after Maple prog. *)

a(n) = coefficient of y in sum_0^n sigma_{n, k}(n-k)!(y-1)^k on y where the sigma_{n, k} have generating function sigma(t, u)=(1-2t^2(u^2)-2t^2(1+t)u^3+3t^4(u^4))(1-tu)^(-1)(1-(1+2t)u-tu^2+t^3(u^3))^(-1). - Barbara Haas Margolius (margolius(AT)math.csuohio.edu) Feb 17 2001

More terms from Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Feb 17 2001

A000470 Number of permutations of an n-sequence discordant with three given permutations (see reference) in n-5 places.

Original entry on oeis.org

13, 72, 595, 4096, 39078, 379760, 4181826, 49916448, 647070333, 9035216428, 135236990388, 2159812592384, 36658601139066, 658942295734944, 12504663617290908, 249823152134646144, 5241223014084306270, 115206851288747267148, 2647678812396326064043

Offset: 5

N. J. A. Sloane

nonn

J. Riordan, Discordant permutations, Scripta Math., 20 (1954), 14-23.
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).

J. Riordan, Discordant permutations, Scripta Math., 20 (1954), 14-23. [Annotated scanned copy]

Cf. A000500, A000492, A000440, A000476, A000380, A000388.

seq(f(n,5), n=5..30); # code for f(n,k) is given in A000440 - Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Feb 17 2001

sigma[t_, u_] = (1 - 2 t^2 (u^2) - 2 t^2 (1 + t) u^3 + 3 t^4 (u^4)) (1 - t* u)^(-1) (1 - (1 + 2 t) u - t *u^2 + t^3 (u^3))^(-1);ds[t_, n_] := D[sigma[t, u], {u, n}] /. u -> 0; su[n_] := su[n] = Sum[ Coefficient[ds[t, n]/n!, t, j]*(n - j)!*(y - 1)^j, {j, 0, n}]; f[n_, k_] := Coefficient[su[n], y, k]; Table[f[n, 5], {n, 5, 23}] (* Jean-François Alcover, Sep 01 2011, after Maple prog. *)

a(n) = coefficient of y^5 in sum_0^n sigma_{n, k}(n - k)!(y - 1)^k on y where the sigma_{n, k} have generating function sigma(t, u) = (1 - 2t^2(u^2) - 2t^2(1 + t)u^3 + 3t^4(u^4))(1 - tu)^(-1)(1 - (1 + 2t)u - tu^2 + t^3(u^3))^(-1). - Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Feb 17 2001

More terms from Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Feb 17 2001

A000492 Number of permutations of an n-sequence discordant with three given permutations (see reference) in n-6 places.

Original entry on oeis.org

20, 154, 1676, 14292, 155690, 1731708, 21264624, 280260864, 3970116255, 60113625680, 969368687752, 16588175089420, 300272980075896, 5733025551810600, 115148956467702600, 2427199940533198992, 53576182138937428377, 1235917889588345408586

Offset: 6

N. J. A. Sloane

nonn

J. Riordan, Discordant permutations, Scripta Math., 20 (1954), 14-23.
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).

J. Riordan, Discordant permutations, Scripta Math., 20 (1954), 14-23. [Annotated scanned copy]

Cf. A000500, A000470, A000440, A000476, A000380, A000388.

seq(f(n,6), n=6..30); # code for f(n,k) is given in A000440 - Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Feb 17 2001

sigma[t_, u_] = (1-2t^2 (u^2) - 2t^2 (1+t) u^3 + 3t^4 (u^4)) (1-t*u)^(-1) (1-(1+2t)u - t*u^2 + t^3 (u^3))^(-1); ds[t_, n_] := D[sigma[t, u], {u, n}] /. u -> 0; f[n_, k_] := Coefficient[Sum[ Coefficient[ ds[t, n]/n!, t, j]*(n-j)!*(y-1)^j, {j, 0, n}], y, k]; a[n_] := f[n, 6]; Table[a[n], {n, 6, 25}] (* Jean-François Alcover, Feb 09 2016 *)

a(n) = coefficient of y^6 in sum_0^n sigma_{n, k}(n - k)!(y - 1)^k on y where the sigma_{n, k} have generating function sigma(t, u) = (1 - 2t^2(u^2) - 2t^2(1 + t)u^3 + 3t^4(u^4))(1 - tu)^(-1)(1 - (1 + 2t)u - tu^2 + t^3(u^3))^(-1). - Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Feb 17 2001

More terms from Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Feb 17 2001

A000500 Number of permutations of an n-sequence discordant with three given permutations (see reference) in n-7 places.

Original entry on oeis.org

31, 304, 4230, 43880, 547338, 6924960, 94714620, 1375878816, 21273204330, 348919244768, 6056244249682, 110955673493568, 2140465858763844, 43379533256972640, 921616584567907176, 20485188316420940640, 475499882089797554181, 11506280235885243825696

Offset: 7

N. J. A. Sloane

nonn

J. Riordan, Discordant permutations, Scripta Math., 20 (1954), 14-23.
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).

J. Riordan, Discordant permutations, Scripta Math., 20 (1954), 14-23. [Annotated scanned copy]

Cf. A000492, A000470, A000440, A000476, A000380, A000388.

seq(f(n,7), n=7..30); # code for f(n,k) is given in A000440 - Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Feb 17 2001

sigma[t_, u_] := (1-2*t^2*u^2 - 2*t^2*(1+t)*u^3 + 3*t^4*u^4)/((1-t*u)*(1-(1 + 2*t)*u - t*u^2 + t^3*u^3)); ds[t_, n_] := D[sigma[t, u], {u, n}] /. u -> 0; f[n_, k_] := Coefficient[ Sum[ Coefficient[ds[t, n]/n!, t, j]*(n-j)!*(y-1)^j, {j, 0, n}], y, k]; a[n_] := f[n, 7]; Table[Print[an = a[n]]; an, {n, 7, 24}] (* Jean-François Alcover, Jan 25 2013, after Maple code *)

a(n) = coefficient of y^7 in sum_0^n sigma_{n, k}(n - k)!(y - 1)^k on y where the sigma_{n, k} have generating function sigma(t, u) = (1 - 2t^2(u^2) - 2t^2(1 + t)u^3 + 3t^4(u^4))(1 - tu)^(-1)(1 - (1 + 2t)u - tu^2 + t^3(u^3))^(-1). - Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Feb 17 2001

More terms from Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Feb 17 2001

A094315 Triangle read by rows giving number of circular permutations of n letters such that all letters are displaced by no more than k places from their original position.

Original entry on oeis.org

1, 0, 1, 0, 0, 2, 0, 0, 0, 6, 1, 0, 6, 8, 9, 2, 15, 20, 40, 30, 13, 20, 72, 180, 176, 180, 72, 20, 144, 609, 1106, 1421, 980, 595, 154, 31, 1265, 4960, 9292, 10352, 8326, 4096, 1676, 304, 49

Offset: 0

N. J. A. Sloane, based on a suggestion from Anthony C Robin, Jun 02 2004

The n-th row sums to n!.

			1;
0, 1;
0, 0, 2;
0, 0, 0, 6;
1, 0, 6, 8, 9;
2, 15, 20, 40, 30, 13;
20, 72, 180, 176, 180, 72, 20;
144, 609, 1106, 1421, 980, 595, 154, 31;

J. Riordan, Discordant permutations, Scripta Math., 20 (1954), 14-23.

Anthony C. Robin, 90.72 Circular Wife Swapping, The Mathematical Gazette, Vol. 90, No. 519 (Nov., 2006), pp. 471-478.
J. Riordan, Discordant permutations, Scripta Math., 20 (1954), 14-23. [Annotated scanned copy] (See Table 2)

Diagonals give A000183 (which has further references), A000476, A000388, A000380, A000440, etc.

cp's OEIS Frontend

A000388 Number of permutations of an n-sequence discordant with three given permutations (see reference) in n-2 places.

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A000380 Number of permutations of an n-sequence discordant with three given permutations (see reference) in n-3 places.

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A000476 Number of permutations of an n-sequence discordant with three given permutations (see reference) in n-1 places.

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A000470 Number of permutations of an n-sequence discordant with three given permutations (see reference) in n-5 places.

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A000492 Number of permutations of an n-sequence discordant with three given permutations (see reference) in n-6 places.

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A000500 Number of permutations of an n-sequence discordant with three given permutations (see reference) in n-7 places.

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A094315 Triangle read by rows giving number of circular permutations of n letters such that all letters are displaced by no more than k places from their original position.

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