A000211 -id:A000211 - cp's OEIS frontend

A000565 Number of discordant permutations.

31, 696, 5823, 29380, 108933, 327840, 848380, 1958004, 4130895, 8107024, 14990889, 26372124, 44470165, 72305160, 113897310, 174496828, 260846703, 381480456, 547057075, 770735316, 1068589557, 1460069392, 1968505152, 2621661540

Offset: 7

N. J. A. Sloane

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).

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
J. Riordan, Discordant permutations, Scripta Math., 20 (1954), 14-23. [Annotated scanned copy]
Index entries for linear recurrences with constant coefficients, signature (8,-28, 56,-70,56,-28,8,-1).

[243/560*n^7-243/16*n^6+3591/16*n^5-28737/16*n^4+ 82257/10*n^3 - 81931/4*n^2+151931/7*n: n in [7..45]]; // Vincenzo Librandi, Feb 10 2016

pp := n - >243/560*n^7 - 243/16*n^6 + 3591/16*n^5 - 28737/16*n^4 + 82257/10*n^3 - 81931/4*n^2 + 151931/7*n; seq(pp(n), n=0..30); # Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Feb 17 2001
A000565:=-(12*z**7-6*z**6-131*z**4+88*z**5-1123*z**2-548*z**3-31-448*z)/(z-1)**8; # conjectured by Simon Plouffe in his 1992 dissertation

LinearRecurrence[{8, -28, 56, -70, 56, -28, 8, -1}, {31, 696, 5823, 29380, 108933, 327840, 848380, 1958004}, 30] (* Jean-François Alcover, Feb 10 2016 *)

From Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Feb 17 2001: (Start)
G.f.: -x^7(12x^7 - 6x^6 + 88x^5 - 131x^4 - 548x^3 - 1123x^2 - 448x - 31) / ((1 - x)^8).
a(n) = 243/560n^7 - 243/16n^6 + 3591/16n^5 - 28737/16n^4 + 82257/10n^3 - 81931/4n^2 + 151931/7n, for n>6. (End)

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

A000562 Number of discordant permutations.

Original entry on oeis.org

9, 95, 420, 1225, 2834, 5652, 10165, 16940, 26625, 39949, 57722, 80835, 110260, 147050, 192339, 247342, 313355, 391755, 484000, 591629, 716262, 859600, 1023425, 1209600, 1420069, 1656857, 1922070, 2217895, 2546600, 2910534, 3312127, 3753890, 4238415, 4768375, 5346524, 5975697

Offset: 4

N. J. A. Sloane

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).

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
J. Riordan, Discordant permutations, Scripta Math., 20 (1954), 14-23. [Annotated scanned copy]
Index entries for linear recurrences with constant coefficients, signature (5,-10, 10,-5,1).

[9] cat [27/8*n^4-135/4*n^3+921/8*n^2-539/4*n: n in [5..45]]; // Vincenzo Librandi, Feb 10 2016

ff := n->27/8*n^4-135/4*n^3+921/8*n^2-539/4*n; seq(ff(n), n=5..40); # Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Feb 17 2001
A000562:=(-9-50*z-35*z**2+15*z**3-4*z**4+2*z**5)/(z-1)**5; # conjectured by Simon Plouffe in his 1992 dissertation

Join[{9}, LinearRecurrence[{5, -10, 10, -5, 1}, {95, 420, 1225, 2834, 5652}, 40]] (* Jean-François Alcover, Feb 10 2016 *)

From Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Feb 17 2001: (Start)
G.f.: -x^4(2x^5 - 4x^4 + 15x^3 - 35x^2 - 50x - 9) / ((1 - x)^5).
a(n) = 27/8n^4 - 135/4n^3 + 921/8n^2 - 539/4n, n>4. (End)

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

A000563 Number of discordant permutations.

Original entry on oeis.org

13, 192, 1085, 3880, 10656, 24626, 50380, 94128, 163943, 270004, 424839, 643568, 944146, 1347606, 1878302, 2564152, 3436881, 4532264, 5890369, 7555800, 9577940, 12011194, 14915232, 18355232, 22402123, 27132828, 32630507, 38984800, 46292070

Offset: 5

N. J. A. Sloane

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).

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
J. Riordan, Discordant permutations, Scripta Math., 20 (1954), 14-23. [Annotated scanned copy]
Index entries for linear recurrences with constant coefficients, signature (6,-15, 20,-15,6,-1).

[81/40*n^5-135/4*n^4+1719/8*n^3-2487/4*n^2+3463/5*n: n in [5..45]]; // Vincenzo Librandi, Feb 10 2016

r := n->81/40*n^5-135/4*n^4+1719/8*n^3-2487/4*n^2+3463/5*n; seq(r(n), n=5..40); # Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Feb 17 2001
A000563:=-(-13-114*z-128*z**2+10*z**3-6*z**4+8*z**5)/(z-1)**6; # conjectured by Simon Plouffe in his 1992 dissertation

LinearRecurrence[{6, -15, 20, -15, 6, -1}, {13, 192, 1085, 3880, 10656, 24626}, 30] (* Jean-François Alcover, Feb 10 2016 *)

From Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Feb 17 2001: (Start)
G.f.: -x^5(8x^5 - 6x^4 + 10x^3 - 128x^2 - 114x - 13) / ((1 - x)^6).
a(n) = 81/40n^5 - 135/4n^4 + 1719/8n^3 - 2487/4n^2 + 3463/5n, n>4. (End)

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

A000564 Number of discordant permutations.

Original entry on oeis.org

20, 371, 2588, 11097, 35645, 94457, 218124, 454220, 872648, 1571715, 2684936, 4388567, 6909867, 10536089, 15624200, 22611330, 32025950, 44499779, 60780420, 81744725, 108412889, 141963273, 183747956, 235309016, 298395540

Offset: 6

N. J. A. Sloane

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).

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
J. Riordan, Discordant permutations, Scripta Math., 20 (1954), 14-23. [Annotated scanned copy]
Index entries for linear recurrences with constant coefficients, signature (7,-21, 35,-35,21,-7,1).

[20] cat [81/80*n^6-405/16*n^5+4113/16*n^4-21267/16*n^3+140357/40*n^2 - 7587/2*n: n in [7..45]]; // Vincenzo Librandi, Feb 10 2016

rr := n - >81/80*n^6 - 405/16*n^5 + 4113/16*n^4 - 21267/16*n^3 + 140357/40*n^2 - 7587/2*n; seq(rr(n), n=7..40); # Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Feb 17 2001
A000564:=(-20-231*z-411*z**2-72*z**3-29*z**4+36*z**5-4*z**6+2*z**7)/(z-1)**7; # conjectured by Simon Plouffe in his 1992 dissertation

Join[{20}, LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {371, 2588, 11097, 35645, 94457, 218124, 454220}, 30]] (* Jean-François Alcover, Feb 10 2016 *)

From Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Feb 17 2001: (Start)
G.f.: -x^6(2x^7 - 4x^6 + 36x^5 - 29x^4 - 72x^3 - 411x^2 - 231x - 20) / ((1 - x)^7).
a(n) = 81/80n^6 - 405/16n^5 + 4113/16n^4 - 21267/16n^3 + 140357/40n^2 - 7587/2n, n>6. (End)

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

A001926 G.f.: (1+x)^2/[(1-x)^4(1-x-x^2)^3].

Original entry on oeis.org

1, 9, 46, 177, 571, 1632, 4270, 10446, 24244, 53942, 115954, 242240, 494087, 987503, 1939634, 3753007, 7167461, 13532608, 25293964, 46856332, 86110792, 157125052, 284866900, 513470464, 920659517, 1642844485, 2918680214, 5164483453, 9104522495, 15995633440

Offset: 0

N. J. A. Sloane

nonn

From rook polynomials.

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).

T. D. Noe, Table of n, a(n) for n = 0..1000
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
J. Riordan, Discordant permutations, Scripta Math., 20 (1954), 14-23. [Annotated scanned copy]
Index entries for linear recurrences with constant coefficients, signature (7,-18,17,7,-24,9,9,-6,-1,1).

Second differences are in A002941.

A001926:=-(1+z)**2/(z**2+z-1)**3/(z-1)**4; # conjectured (correctly) by Simon Plouffe in his 1992 dissertation

nn = 30; CoefficientList[Series[(1 + x)^2/((1 - x)^4 (1 - x - x^2)^3), {x, 0, nn}], x] (* T. D. Noe, Aug 17 2012 *)
LinearRecurrence[{7,-18,17,7,-24,9,9,-6,-1,1},{1,9,46,177,571,1632,4270,10446,24244,53942},30] (* Harvey P. Dale, Apr 30 2022 *)

Edited by N. J. A. Sloane, Apr 10 2009

A259454 Triangle T(n,k) (0 <= k <= n) read by rows, arising from the study of rook polynomials.

Original entry on oeis.org

1, 1, 3, 1, 6, 7, 1, 9, 22, 14, 1, 12, 46, 64, 26, 1, 15, 79, 177, 162, 46, 1, 18, 121, 380, 571, 374, 79, 1, 21, 172, 700, 1496, 1632, 809, 133, 1, 24, 232, 1164, 3261, 5116, 4270, 1668, 221, 1, 27, 301, 1799, 6271, 13013, 15754, 10446, 3316, 364

Offset: 0

N. J. A. Sloane, Jun 28 2015

See Riordan 1954 page 18 equation (9). - Michael Somos, Aug 26 2015

			Triangle T(n,k) begins:
1;
1,  3;
1,  6,  7;
1,  9,  22,   14;
1, 12,  46,   64,   26;
1, 15,  79,  177,  162,   46;
1, 18, 121,  380,  571,  374,   79;
1, 21, 172,  700, 1496, 1632,  809,  133;
1, 24, 232, 1164, 3261, 5116, 4270, 1668, 221;
G.f. = 1 + (1 + 3*t)*u + (1 + 6*t + 7*t^2)*u^2 + ...

Alois P. Heinz, Rows n = 0..140, flattened
J. Riordan, Discordant permutations, Scripta Math., 20 (1954), 14-23. [Annotated scanned copy] (See triangle on page 18)

Some diagonals: A001924, A001925, A001926.

T:= proc(n, k) option remember; `if`(k<0 or k>n, 0,
      T(n-1, k) +2*T(n-1, k-1) +T(n-2, k-1)
     -T(n-3, k-3) +`if`(n=k, 1, 0))
    end:
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Jul 02 2015

T[n_, k_] /; 0 <= k <= n := T[n, k] = T[n-1, k] + 2*T[n-1, k-1] + T[n-2, k - 1] - T[n-3, k-3] + Boole[n == k]; T[, ] = 0; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 18 2016 *)

{T(n, k) = polcoeff( polcoeff( 1 / ((1 - y*x) * (1 - (1 + 2*y)*x - y*x^2 + y^3*x^3)) + x * O(x^n), n), k)}; /* Michael Somos, Aug 26 2015 */

From Eq. (11) of Riordan (1954): T(n,k) = T(n-1,k) + 2*T(n-1,k-1) + T(n-2,k-1) - T(n-3,k-3) + delta(n,k), where delta(n,k)=1 iff n=k, otherwise 0.

Sum_{n, k} T(n, k) * x^n*y^k = 1 / ((1 - y*x) * (1 - (1 + 2*y)*x - y*x^2 + y^3*x^3)). - Michael Somos, Aug 26 2015

More terms from Alois P. Heinz, Jul 02 2015

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.

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A000565 Number of discordant permutations.

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A000562 Number of discordant permutations.

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A000563 Number of discordant permutations.

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A000564 Number of discordant permutations.

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Magma

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A001926 G.f.: (1+x)^2/[(1-x)^4(1-x-x^2)^3].

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A259454 Triangle T(n,k) (0 <= k <= n) read by rows, arising from the study of rook polynomials.

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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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