A037223 -id:A037223 - cp's OEIS frontend

A000899 Number of solutions to the rook problem on an n X n board having a certain symmetry group (see Robinson for details).

0, 0, 0, 1, 9, 70, 571, 4820, 44676, 450824, 4980274, 59834748, 778230060, 10896609768, 163456629604, 2615335902176, 44460874280032, 800296440705472, 15205636325496568, 304112744618157872, 6386367741011250672

Offset: 1

N. J. A. Sloane

L. C. Larson, The number of essentially different nonattacking rook arrangements, J. Recreat. Math., 7 (No. 3, 1974), circa pages 180-181.
R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976).
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 = 1..200
L. C. Larson, The number of essentially different nonattacking rook arrangements, J. Recreat. Math., 7 (No. 3, 1974), circa pages 180-181. [Annotated scan of pages 180 and 181 only]
E. Lucas, Théorie des Nombres, Gauthier-Villars, Paris, 1891, Vol. 1, p. 222.
E. Lucas, Théorie des nombres (annotated scans of a few selected pages)
R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976). (Annotated scanned copy)

Cf. A000900.

Maple
```
For Maple program see A000903.
```

a[n_] := ((n+1)! - (2*Floor[(n+1)/2])!! - 2*Sum[Binomial[n+1, 2*k]*(2*k-1)!!, {k, 0, (n+1)/2}] + 2*Sum[2^k*BellB[k]*StirlingS1[Floor[(n+1)/2], k], {k, 0, Floor[(n+1)/2]}])/8; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Dec 23 2013, from explicit formulas *)

a(n)=(A000142(n)-2*A000085(n)-A037223(n)+2*A000898(floor(n/2)))/8 (all of which have explicit formulas).

For asymptotics see the Robinson paper.

More terms from Vladeta Jovovic, May 09 2000

A000903 Number of inequivalent ways of placing n nonattacking rooks on n X n board up to rotations and reflections of the board.

Original entry on oeis.org

1, 1, 2, 7, 23, 115, 694, 5282, 46066, 456454, 4999004, 59916028, 778525516, 10897964660, 163461964024, 2615361578344, 44460982752488, 800296985768776, 15205638776753680, 304112757426239984, 6386367801916347184

Offset: 1

N. J. A. Sloane

			For n=4 the 7 solutions may be taken to be 1234,1243,1324,1423,1432,2143,2413.

L. C. Larson, The number of essentially different nonattacking rook arrangements, J. Recreat. Math., 7 (No. 3, 1974), circa pages 180-181.
R. C. Read, personal communication.
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).
Z. Stankova and J. West, A new class of Wilf-equivalent permutations, J. Algeb. Combin., 15 (2002), 271-290.

N. J. A. Sloane, Table of n, a(n) for n = 1..100
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
L. C. Larson, The number of essentially different nonattacking rook arrangements, J. Recreat. Math., 7 (No. 3, 1974), circa pages 180-181. [Annotated scan of pages 180 and 181 only]
E. Lucas, Théorie des Nombres, Gauthier-Villars, Paris, 1891, Vol. 1, p. 222.
E. Lucas, Théorie des nombres (annotated scans of a few selected pages)
R. C. Read, Letter to N. J. A. Sloane, Oct. 29, 1976
R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976).
R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976). (Annotated scanned copy)
Zvezdelina Stankova-Frenkel and Julian West, A new class of Wilf-equivalent permutations, arXiv:math/0103152 [math.CO], 2001. See Fig. 9.
Eric Weisstein's World of Mathematics, Rook Number.
Eric Weisstein's World of Mathematics, Rooks Problem.

Cf. A000142, A037223, A037224, A000085, A005635, A099952, A263685.

Maple programs for A000142, A037223, A122670, A001813, A000085, A000898, A000407, A000902, A000900, A000901, A000899, A000903
P:=n->n!; # Gives A000142
G:=proc(n) local k; k:=floor(n/2); k!*2^k; end; # Gives A037223, A000165
R:=proc(n) local m; if n mod 4 = 2 or n mod 4 = 3 then RETURN(0); fi; m:=floor(n/4); (2*m)!/m!; end; # Gives A122670, A001813
unprotect(D); D:=proc(n) option remember; if n <= 1 then 1 else D(n-1)+(n-1)*D(n-2); fi; end; # Gives A000085
B:=proc(n) option remember; if n <= 1 then RETURN(1); fi; if n mod 2 = 1 then RETURN(B(n-1)); fi; 2*B(n-2) + (n-2)*B(n-4); end; # Gives A000898 (doubled up)
rho:=n->R(n)/2; # Gives A000407, aerated
beta:=n->B(n)/2; # Gives A000902, doubled up
delta:=n->(D(n)-B(n))/2; # Gives A000900
unprotect(gamma); gamma:=n-> if n <= 1 then RETURN(0) else (G(n)-B(n)-R(n))/4; fi; # Gives A000901, doubled up
alpha:=n->P(n)/8-G(n)/8+B(n)/4-D(n)/4; # Gives A000899
unprotect(sigma); sigma:=n-> if n <= 1 then RETURN(1); else P(n)/8+G(n)/8+R(n)/4+D(n)/4; fi; #Gives A000903

c[n_] := Floor[n/2]! 2^Floor[n/2];
r[n_] := If[Mod[n, 4] > 1, 0, m = Floor[n/4]; If[m == 0, 1, (2 m)!/m!]];
d[0] = d[1] = 1; d[n_] := d[n] = (n - 1)d[n - 2] + d[n - 1];
a[1] = 1; a[n_] := (n! + c[n] + 2 r[n] + 2 d[n])/8;
Array[a, 21] (* Jean-François Alcover, Apr 06 2011, after Matthias Engelhardt, further improved by Robert G. Wilson v *)

If n>1 then a(n) = 1/8 * (F(n) + C(n) + 2 * R(n) + 2 * D(n)), where F(n) = A000142(n) [all solutions, i.e., factorials], C(n) = A037223(n) [central symmetric solutions], R(n) = A037224(n) [rotationally symmetric solutions] and D(n) = A000085(n) [symmetric solutions by reflection at a diagonal]. - Matthias Engelhardt, Apr 05 2000

For asymptotics see the Robinson paper.

More terms from David W. Wilson, Jul 13 2003

A037224 Number of permutations p of {1,2,3...,n} that are fixed points under the operation of first reversing p, then taking the inverse.

Original entry on oeis.org

1, 0, 0, 2, 2, 0, 0, 12, 12, 0, 0, 120, 120, 0, 0, 1680, 1680, 0, 0, 30240, 30240, 0, 0, 665280, 665280, 0, 0, 17297280, 17297280, 0, 0, 518918400, 518918400, 0, 0, 17643225600, 17643225600, 0, 0, 670442572800, 670442572800, 0, 0, 28158588057600

Offset: 1

Miklos SZABO (mike(AT)ludens.elte.hu)

A122670 is an essentially identical sequence.
Also the number of rotationally symmetric solutions to non-attacking rooks problem on an n X n board.
Reversal of a permutation reflects the associated permutation matrix through an axis parallel to its sides, while inversion reflects the matrix through its main diagonal. The composition of these two operations is rotation by 90 degrees, and so permutations fixed by this composition correspond to rotationally symmetric rook diagrams by taking the associated permutation matrix. [Ian Duff, Mar 09 2007 and Joel B. Lewis, Jun 10 2009]
Equivalently, the number of permutations fixed by first inverting and then reversing. We may also replace "reversing" with "complementing" in the preceding sentences, where the complement of (w(1), ..., w(n)) is (n + 1 - w(1), ..., n + 1 - w(n)). [Joel B. Lewis, Jun 10 2009]

			Let p be the permutation {11,1,9,3,7,5,8,6,10,4,12,2} of {1,2,3,..,12}. Then the reverse Rp of p is {2,12,4,10,6,8,5,7,3,9,1,11} and the inverse IRp of Rp is {11,1,9,3,7,5,8,6,10,4,12,2}. Thus p counts as one of the a(12)=120 fixed-points for n=12.

C. Bebeacua, T. Mansour, A. Postnikov and S. Severini, On the X-rays of permutations, arXiv:math/0506334 [math.CO], 2005.
M. Szabo, Non-attacking Queens Problem Page

Cf. A001813, A033148, A032522, A037223, A122670.

a:= n-> `if`(irem(n, 4, 'm')>1, 0,
        `if`(m=0, 1, (2*m-1)! * 2/(m-1)!)):
seq(a(n), n=1..99);  # Alois P. Heinz, Jan 21 2011

{1}~Join~Table[If[MemberQ[{0, 1}, Mod[n, 4]], (2 # - 1)!*2/(# - 1)! &[Floor[n/4]], 0], {n, 2, 44}] (* Michael De Vlieger, Oct 05 2016 *)

a(n)=
{
    if ( n%4>=2, return(0) );
    n = n\4;
    if ( n==0, return(1) );
    return( (2*n-1)!*2/(n-1)! );
}
vector(55,n,a(n)) /* Joerg Arndt, Jan 21 2011 */

a(4n) = a(4n+1) = (2n-1)!*2/(n-1)!, a(4n+2) = a(4n+3) = 0.

Edited by N. J. A. Sloane, Jun 12 2009, incorporating comments from John W. Layman, Sep 17 2004

A055634 2-adic factorial function.

Original entry on oeis.org

1, -1, 1, -3, 3, -15, 15, -105, 105, -945, 945, -10395, 10395, -135135, 135135, -2027025, 2027025, -34459425, 34459425, -654729075, 654729075, -13749310575, 13749310575, -316234143225, 316234143225, -7905853580625, 7905853580625, -213458046676875, 213458046676875

Offset: 0

Michael Somos, Jun 06 2000

sign

Also known as Morita's 2-adic gamma function. - Harry Richman, Jul 26 2023

Serge Lang, Cyclotomic Fields I and II, Springer-Verlag, 1990, p. 315.

Kenny Lau, Table of n, a(n) for n = 0..806
Daniel Barsky, On Morita's p-adic Gamma function, Groupe de travail d'analyse ultramétrique, 5 (1977-1978), Talk no. 3, 6 p.
Wikipedia, P-adic gamma function.

Cf. A000142, A006882, A001147, A133221.

/* Based on Gauss factorial n_2!: */ k:=2; [IsZero(n) select 1 else (-1)^n*&*[j: j in [1..n] | IsOne(GCD(j,k))]: n in [0..30]]; // Bruno Berselli, Dec 10 2013

a[ n_] := If[ n < 0, 0, n! (-1)^n / (n - Mod[n, 2])!!]; (* Michael Somos, Jun 30 2018 *)
4[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ (1 - x) Exp[x^2/2], {x, 0, n}]]; (* Michael Somos, Jun 30 2018 *)

{a(n) = if( n<1, 1, -if( n%2, n * a(n-1), a(n-1)))};

a(n)=(-1)^n*(n=bitor(n-1,1))!/(n\2)!>>(n\2) \\ Charles R Greathouse IV, Oct 01 2012

def Gauss_factorial(N, n): return mul(j for j in (1..N) if gcd(j, n) == 1)
def A055634(n): return (-1)^n*Gauss_factorial(n, 2)
[A055634(n) for n in (0..28)]  # Peter Luschny, Oct 01 2012

a(2*n) = -a(2*n - 1) = (2*n - 1)!!

a(n) = (-1)^n*n!/A037223(n), A037223(n) = 2^floor(n/2)*floor(n/2)!. Exponential generating function: (1-x)*exp(x^2/2). - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Nov 01 2002

A033148 Number of rotationally symmetric solutions for queens on n X n board.

Original entry on oeis.org

1, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 8, 8, 0, 0, 64, 128, 0, 0, 480, 704, 0, 0, 3328, 3264, 0, 0, 32896, 43776, 0, 0, 406784, 667904, 0, 0, 5845504, 8650752, 0, 0, 77184000, 101492736, 0, 0, 1261588480, 1795233792, 0, 0, 21517426688, 35028172800, 0, 0, 406875119616, 652044443648, 0, 0, 8613581094912, 12530550128640, 0, 0, 194409626533888, 291826098503680, 0, 0

Offset: 1

Miklos SZABO (mike(AT)ludens.elte.hu)

From Don Knuth, Jul 17 2015: (Start)
Ahrens proved that a(n)=0 unless n=4k or 4k+1. He also proved that in the latter case, a(n) is a multiple of 2^k. He found all solutions when n was less than 20.
Kraitchik carried the calculations further (for n less than 28). In his book he tabulated only the values a(n)/2^k. He had correct entries for n=21 and n=25, but his values for n=20 and n=24 were 1 too small -- of course he had calculated everything by hand! (End)

W. Ahrens, Mathematische Unterhaltungen und Spiele, 2nd edition, volume 1, Teubner, 1910, pages 249-258.
Maurice Kraitchik, Le problème des reines, Bruxelles: L'Échiquier, 1926, page 18.

Tricia M. Brown, Kaleidoscopes, Chessboards, and Symmetry, Journal of Humanistic Mathematics, Volume 6 Issue 1 ( January 2016), pages 110-126.
P. Capstick and K. McCann, The problem of the n queens, apparently unpublished, no date (circa 1990?) [Scanned copy]
Gheorghe Coserea, Solutions for n=20.
Gheorghe Coserea, Solutions for n=24.
Gheorghe Coserea, MiniZinc model for generating solutions.
YuhPyng Shieh, Cyclic Complete Mappings Counting Problems
M. Szabo, Non-attacking Queens Problem Page

Cf. A002562, A032522, A037223, A037224, A260189.

More terms from Jieh Hsiang and YuhPyng Shieh (arping(AT)turing.csie.ntu.edu.tw), May 20 2002

A032522 Number of point symmetric solutions to non-attacking queens problem on n X n board.

Original entry on oeis.org

1, 0, 0, 2, 2, 4, 8, 4, 16, 12, 48, 80, 136, 420, 1240, 3000, 8152, 18104, 44184, 144620, 375664, 1250692, 3581240, 11675080, 34132592, 115718268, 320403024, 1250901440, 3600075088, 14589438024, 43266334696, 181254386312

Offset: 1

Miklos SZABO (mike(AT)ludens.elte.hu)

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).
R. J. Walker, An enumerative technique for a class of combinatorial problems, pp. 91-94 of Proc. Sympos. Applied Math., vol. 10, Amer. Math. Soc., 1960.

W. Schubert, Table of n, a(n) for n = 1..40
Tricia M. Brown, Kaleidoscopes, Chessboards, and Symmetry, Journal of Humanistic Mathematics, Volume 6 Issue 1 ( January 2016), pages 110-126.
Gheorghe Coserea, Solutions for n=10.
Gheorghe Coserea, Solutions for n=11.
Gheorghe Coserea, MiniZinc model for generating solutions.
W. Schubert, N-Queens page
M. Szabo, Non-attacking Queens Problem Page

Cf. A002562, A033148, A037224, A037223.

More terms for n = 33..36 from W. Schubert, Jul 31 2009

A249130 Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.

Original entry on oeis.org

1, 2, 1, 2, 2, 1, 8, 6, 2, 1, 8, 16, 10, 2, 1, 48, 44, 28, 16, 2, 1, 48, 144, 104, 40, 22, 2, 1, 384, 400, 368, 232, 56, 30, 2, 1, 384, 1536, 1232, 688, 408, 72, 38, 2, 1, 3840, 4384, 5216, 3552, 1248, 708, 92, 48, 2, 1, 3840, 19200, 16704, 12096, 7632, 1968, 1088, 112, 58, 2, 1

Offset: 0

Clark Kimberling, Oct 22 2014

The polynomial p(n,x) is the numerator of the rational function given by f(n,x) = x + 2*floor((n+1)/2)/f(n-1,x), where f(0,x) = 1.
(Sum of numbers in row n) = A249131(n) for n >= 0.
(Column 1) = A037223.

			f(0,x) = 1/1, so that p(0,x) = 1;
f(1,x) = (2 + x)/1, so that p(1,x) = 2 + x;
f(2,x) = (2 + 2*x + x^2)/(3 + x), so that p(2,x) = 2 + 2*x + x^2.
First 6 rows of the triangle of coefficients:
  1
  2    1
  2    2    1
  8    6    2    1
  8    16   10   2    1
  48   44   28   16   2   1

Clark Kimberling, Rows 0..100, flattened

Cf. A249131, A037223, A249128.

z = 15; p[x_, n_] := x + 2 Floor[n/2]/p[x, n - 1]; p[x_, 1] = 1;
t = Table[Factor[p[x, n]], {n, 1, z}]
u = Numerator[t]
TableForm[Table[CoefficientList[u[[n]], x], {n, 1, z}]] (* A249130 array *)
Flatten[CoefficientList[u, x]] (* A249130 sequence *)

A076051 Sum of product of odd numbers <= n and the product of even numbers <= n.

Original entry on oeis.org

2, 3, 5, 11, 23, 63, 153, 489, 1329, 4785, 14235, 56475, 181215, 780255, 2672145, 12348945, 44781345, 220253985, 840523635, 4370620275, 17465201775, 95498916975, 397983749625, 2278224696825, 9867844134225, 58917607974225

Offset: 1

Emrehan Halici (emrehan(AT)halici.com.tr), Oct 30 2002

G. C. Greubel, Table of n, a(n) for n = 1..800

Cf. A037223, A055634, A060696.

A037223[n_] := 2^(Floor[n/2])*(Floor[n/2])!; Table[A037223[n] + n!/A037223[n] , {n,1,50}] (* G. C. Greubel, May 23 2017 *)
With[{nn = 25}, CoefficientList[Series[1 + x + (1 + x + x^2) *(Exp[x^2/2] *(1 + Sqrt[Pi/2]*Erf[x/Sqrt[2]])), {x, 0, nn}], x] Range[0, nn]!] (* G. C. Greubel, May 25 2017 *)

for(n=1, 50, print1(2^(floor(n/2))*(floor(n/2))! + n!/(2^(floor(n/2))*(floor(n/2))!), ", ")) \\ G. C. Greubel, May 23 2017

a(n) = o(n)+ e(n) where; o(n)=the product of odd numbers from 1 to n e(n)=the product of even numbers from 2 to n.
From Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Nov 01 2002: (Start)
a(n) = A060696(n+1).
a(n) = A037223(n) + abs(A055634(n)).
a(n) = A037223(n) + n! / A037223(n), where A037223(n) = 2^floor(n/2) * floor(n/2)!, for n>=2.
a(1)=2, a(2)=3, a(3)=5, a(n) = (n-1)*a(n-2) + (n-2)!! for n >= 4.
E.g.f.: 1 + x + (1+x+x^2)*(exp(x^2/2)*(1+sqrt(Pi/2)*erf(x/sqrt(2)))), where erf denotes the error function. (End)

More terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Nov 01 2002

a(1) corrected by G. C. Greubel, May 23 2017

A155517 Triangle read by rows: T(n,k) is the number of permutations p of {1,2,...,n} for which the number of j < ceiling(n/2) such that p(j) + p(n+1-j) = n+1 is equal to k (n>=1; 0<=k <=ceiling(n/2)).

Original entry on oeis.org

0, 1, 0, 2, 4, 0, 2, 16, 0, 8, 64, 48, 0, 8, 384, 288, 0, 48, 2880, 1536, 576, 0, 48, 23040, 12288, 4608, 0, 384, 208896, 115200, 30720, 7680, 0, 384, 2088960, 1152000, 307200, 76800, 0, 3840, 23193600, 12533760, 3456000, 614400, 115200, 0, 3840, 278323200

Offset: 1

Emeric Deutsch, Jan 26 2009

For the permutation 31756284 of S_8 we have k=2 because p(2) + p(7) = 1+8 = 9 and p(3) + p(6) = 7+2 = 9; for the permutation 3214756 of S_7 we have k=2 because p(3) + p(5) = 1+7 = 8 and p(4) + p(4) = 4+4 = 8.
Row sums are the factorial numbers (A000142).
Row n contains 1 + ceiling(n/2)entries.
T(2n,n) = n!*2^n = A037223(2n) = number of centrosymmetric permutations in S[2n];
T(2n+1,n+1) = n!*2^n = A037223(2n+1) = number of centrosymmetric permutations in S[2n+1].
T(n,0) = A155518(n).
Sum_{k=0..ceiling(n/2)} k*T(n,k) = A155519(n).

			T(4,2)=8 because we have 1234, 4231, 1324, 4321, 2143, 3142, 2413 and 3412.
Triangle starts:
    0,   1;
    0,   2;
    4,   0,   2;
   16,   0,   8;
   64,  48,   0,   8;
  384, 288,   0,  48;

Cf. A000142, A037223, A055140, A155518, A155519.

g[0] := 1: g[1] := 0: for n from 2 to 20 do g[n] := (2*(n-1))*(g[n-1]+g[n-2]) end do: T := proc (n, k) if `mod`(n, 2) = 0 then 2^((1/2)*n)*factorial((1/2)*n)*g[(1/2)*n-k]*binomial((1/2)*n, k) else 2^((1/2)*n-1/2)*factorial((1/2)*n-1/2)*g[(1/2)*n+1/2-k]*binomial((1/2)*n+1/2, k) end if end proc: for n to 12 do seq(T(n, k), k = 0 .. ceil((1/2)*n)) end do;

T(2n,k) = n!*2^n*A055140(n,k);
T(2n-1,k) = (n-1)!*2^(n-1)*A055140(n,k);
here A055140(n,k) = A053871(n-k)*binomial(n,k), where g(n) = A053871(n) is defined by g(0)=1, g(1)=0, g(n) = 2(n-1)(g(n-1)+g(n-2)).

A198890 Irregular triangle read by rows: row n gives expansion of g.f. for descending plane partitions of order n with no special parts and weight equal to sum of the parts.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 2, 2, 2, 3, 2, 4, 3, 4, 4, 4, 5, 4, 5, 5, 4, 6, 4, 5, 5, 4, 5, 4, 4, 4, 3, 4, 2, 3, 2, 2, 2, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 2, 3, 2, 4, 3, 5, 5, 7, 6, 8, 8, 9, 10, 12, 10, 14, 12, 14, 15, 16, 15, 18, 16, 18, 18, 20, 17, 21, 18, 20, 20, 20, 18, 21, 17, 20, 18, 18, 16, 18, 15, 16, 15, 14, 12, 14, 10, 12, 10, 9, 8, 8, 6, 7, 5, 5, 3, 4, 2, 3, 2, 1, 1, 1, 0, 1

Offset: 1

N. J. A. Sloane, Oct 30 2011

			Rows 1 through 5 are
  1
  1, 0, 1
  1, 0, 1, 1, 0, 1, 1, 0, 1
  1, 0, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 0, 1
  1, 0, 1, 1, 1, 2, 2, 2, 3, 2, 4, 3, 4, 4, 4, 5, 4, 5, 5, 4, 6, 4, 5, 5, 4, 5, 4, 4, 4, 3, 4, 2, 3, 2, 2, 2, 1, 1, 1, 0, 1
From _Peter Bala_, May 29 2022: (Start)
Row 3 generating polynomial:
   Permutation p    Pairs (p(i),p(j)) with p(i) > p(j)       inv_1(p)
       123                       -                              0
       132                     (3,2)                            3
       213                     (2,1)                            2
       231                 (2,1), (3,1)                         5
       312                 (3,1), (3,2)                         6
       321              (3,2), (3,1), (2,1)                     8
Hence R(3,x) = x^0 + x^2 + x^3 + x^5 + x^6 + x^8 = (1 + x^2)*(1 + x^3 + x^6) = ((1 - x^4)/(1 - x^2)) * (1 - x^9)/(1 - x^3). (End)

Peter Bala, A note on A198890
J. Striker, A direct bijection between descending plane partitions with no special parts and permutation matrices, arXiv:1002.3391 [math.CO], 2010-2012.
J. Striker, A direct bijection between descending plane partitions with no special parts and permutation matrices, Discrete Math., 311 (2011).

Row sums give A000142 (factorial numbers).

Cf. A001754, A008302, A037223, A087153.

s:=(k,q)->add(q^i,i=0..k-1);
f:=n->mul(s(i,q^i),i=1..n);
g:=n->seriestolist(series(f(n),q,1000));
for n from 1 to 10 do lprint(g(n)); od:
# alternative program
T := proc (n, k) option remember;
if n = 0 or n = 1 and k = 0 then 1
elif k > ((1/3)*n-1/3)*n*(n+1) then 0
elif k < 0 then 0
else T(n, k-n) + T(n-1, k) - T(n-1, k-n^2) fi end:
seq(print(seq(T(n, k), k = 0..(1/3)*(n-1)*n*(n+1))), n = 1..6); # Peter Bala, Jun 07 2022

From Peter Bala, May 29 2022: (Start)
T(0, 0) = 1; T(1, 0) = 1.
T(n, k) = 0 for k < 0 or k > (1/3)*(n+1)*n*(n-1).
T(n, k) = Sum_{j = 0..n-1} T(n-1, k-n*j); T(n, k) = T(n, k-n) + T(n-1, k) - T(n-1, k-n^2).
T(n,k) = T(n, (1/3)*(n+1)*n*(n-1) - k).
Sum_{k = 0..(1/3)*(n+1)*n*(n-1)} T(n, k) = n!.
Sum_{k = 0..(1/3)*(n+1)*n*(n-1)} (-1)^k*T(n, k) = A037223(n).
Sum_{k = 0..(1/3)*(n+1)*n*(n-1)} k*T(n, k) = (1/3)*n!*binomial(n-1,2) = 2*A001754(n) for n >= 1.
n-th row polynomial R(n,x) = Product_{j = 1..n} (1 - x^(j^2))/(1 - x^j).
let k be a nonnegative integer. Let p = p(1)p(2)...p(n) be a permutation of {1,2,...,n}. We define the k-th inversion number of p by inv_k(p) = Sum_{pairs (i,j), 1 <= i < j <= n, such that p(i) > p(j)} (p(i))^k. The n-th row polynomial R(n,x) equals Sum_{permutations p of {1,2,...,n} } x^(inv_1(p)). An example is given below. For the case k = 0 see A008302.
The x-adic limit of R(n,x) as n -> 00 is the g.f. of A087153. (End)

Name clarified by Ludovic Schwob, Jun 15 2023

cp's OEIS Frontend

A000899 Number of solutions to the rook problem on an n X n board having a certain symmetry group (see Robinson for details).

Views

Author

Keywords

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A000903 Number of inequivalent ways of placing n nonattacking rooks on n X n board up to rotations and reflections of the board.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A037224 Number of permutations p of {1,2,3...,n} that are fixed points under the operation of first reversing p, then taking the inverse.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A055634 2-adic factorial function.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Sage

Formula

A033148 Number of rotationally symmetric solutions for queens on n X n board.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Extensions

A032522 Number of point symmetric solutions to non-attacking queens problem on n X n board.

Views

Author

Keywords

References

Links

Crossrefs

Extensions

A249130 Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs