A037223 -id:A037223 - cp's OEIS frontend

A133922 a(n) is number of permutations (p(1),p(2),p(3),...,p(n)) of (1,2,3,...,n) such that p(k) is coprime to p(n+1-k) for k = all positive integers <= n.

1, 2, 2, 16, 16, 192, 192, 6912, 4608, 230400, 230400, 11612160, 11612160, 1199923200, 588349440, 98594979840, 98594979840, 11076328488960, 11076328488960, 2102897147904000, 1076597725593600, 331238941183180800, 331238941183180800, 66325953940291584000, 56326771107377971200

Offset: 1

Leroy Quet, Jan 07 2008

For n = odd integer the middle term of all counted permutations must be 1.
From Robert Israel, Sep 12 2016: (Start)
Consider the graph with vertices [1,...,n] if n is even, [2,...,n] if n is odd, and edges joining coprime integers.
a(n) is A037223(n) times the number of perfect matchings in this graph.
If n is even, a(n) = A037223(n)*A009679(n/2).
If n is an odd prime, a(n) = a(n-1). (End)

			For n = 6, the permutation (3,2,1,6,4,5) is not counted because p(2)=2 is not coprime to p(5)=4. However, the permutation (3,6,1,4,5,2) is counted because GCD(3,2) = GCD(6,5) = GCD(1,4) = 1.

Robert Israel, Table of n, a(n) for n = 1..31

Cf. A009679, A081123, A037223.

M:= proc(A) option remember;
    local n,t,i,Ai,Ap,inds,isrt,As;
    n:= nops(A);
    if n = 0 then return 1 fi;
    t:= 0;
    for i in A[1] do
      inds:= [$2..i-1,$i+1..n];
      Ai:= subs([1=NULL,i=NULL,seq(inds[i]=i,i=1..n-2)],A[inds]);
      isrt:= sort([$1..n-2],(r,s) -> nops(Ai(r)) <= nops(Ai(s)),output=permutation);
      Ai:= subs([seq(isrt[i]=i,i=1..n-2)],Ai[isrt]);
      t:= t + procname(Ai);
    od;
    t;
end proc:
F:= proc(n) local A;
  if n::odd then
    if isprime(n) then return procname(n-1) fi;
    A:= [seq(select(t -> igcd(t+1,i+1)=1, [$1..i-1,$i+1..n-1]),i=1..n-1)];
  else
    A:= [seq(select(t -> igcd(t,i)=1,[$1..i-1,$i+1..n]),i=1..n)]
  fi;
  M(A)*floor(n/2)!*2^floor(n/2)
end proc;
seq(F(n),n=1..27); # Robert Israel, Sep 12 2016

a(6)-a(15) from Sean A. Irvine, May 17 2010

a(16)-a(25) from Robert Israel, Sep 12 2016

A260189 a(n) = A033148(n) / 2^floor(n/4).

Original entry on oeis.org

1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 4, 8, 0, 0, 15, 22, 0, 0, 52, 51, 0, 0, 257, 342, 0, 0, 1589, 2609, 0, 0, 11417, 16896, 0, 0, 75375, 99114, 0, 0, 616010, 876579, 0, 0, 5253278, 8551800, 0, 0, 49667373, 79595269, 0, 0, 525731268, 764804085, 0, 0, 5932910966, 8905825760, 0, 0

Offset: 1

Vaclav Kotesovec, following a suggestion of Don Knuth, Jul 18 2015

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

P. Capstick and K. McCann, The problem of the n queens, apparently unpublished, no date (circa 1990?) [Scanned copy]

Cf. A002562, A032522, A033148, A037223, A037224.

A277083 Irregular triangle read by rows: T(n,k) = number of size k subsets of S_n that remain unchanged by a rotation of 180 degrees.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 4, 3, 2, 1, 1, 8, 36, 120, 322, 728, 1428, 2472, 3823, 5328, 6728, 7728, 8092, 7728, 6728, 5328, 3823, 2472, 1428, 728, 322, 120, 36, 8, 1, 1, 8, 84, 504, 3178, 15512, 74788, 311144, 1252819, 4577328, 16087512, 52691408, 165911284

Offset: 0

Christian Bean, Sep 28 2016

A permutation, p, can be thought of as a set of points (i, p(i)). If you plot all the points and rotate the picture by 180 degrees then you get a permutation back.

T(n,k) is the number of size k subsets of S_n that remain unchanged by a rotation of 180 degrees.

			For n = 3 and k = 3, the subsets unchanged by rotating 180 degrees are {213,132,123}, {231,312,123}, {321,132,213} and {321,231,312} so T(3,3) = 4.
Triangle starts:
1, 1;
1, 1;
1, 2, 1;
1, 2, 3, 4, 3, 2, 1;

Row lengths give A038507.

Cf. A037223.

T(n,k) = Sum_( binomial( n! - R(n), i ) * binomial( R(n), k-2*i ) for i in [0..floor(k/2)] ) where R(n) = A037223(n).

A277085 Irregular triangle read by rows: T(n,k) = number of size k subsets of S_n that remain unchanged by a rotation of 90 degrees.

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 2, 4, 6, 10, 14, 20, 26, 31, 36, 40, 44, 44, 44, 40, 36, 31, 26, 20, 14, 10, 6, 4, 2, 1, 1, 2, 4, 6, 34, 62, 116, 170, 547, 924, 1624, 2324, 5572, 8820, 14616, 20412, 40509, 60606, 95004, 129402, 224406, 319410

Offset: 0

Christian Bean, Sep 28 2016

A permutation, p, can be thought of as a set of points (i, p(i)). If you plot all the points and rotate the picture by 90 degrees then you get a permutation back.

T(n,k) is the number of size k subsets that remain unchanged by a rotation of 90 degrees.

			For n = 4 and k = 2, the subsets unchanged by a 90-degree rotation are {4321,1234}, {4231,1324}, {3412,2143} and {3142,2413}. Hence T(4,2) = 4.
Triangle starts:
1, 1;
1, 1;
1, 0, 1;
1, 0, 1, 0, 1, 0, 1;

Row lengths give A038507.

Cf. A037223, A037224.

T(n,k) = Sum_( C( R(n) - T(n), i ) * Sum_(C(n! - R(n), j) * C(T(n), k - 4*i - 2*j) for j in [0..floor((k-4*i)/2)] for i in [0..floor(k/4)] ) where R(n) = A037223(n) and T(n) = A037224(n).

A383370 Number of partial orders on {1,2,...,n} that are contained in the usual linear order, whose dual is given by the relabelling k -> n+1-k.

Original entry on oeis.org

1, 1, 2, 3, 12, 25, 172, 482, 5318, 19675, 333768, 1609846, 40832554, 254370640, 9459449890, 75546875426, 4061670272088

Offset: 0

Ludovic Schwob, Apr 24 2025

a(n) is the number of n X n upper triangular Boolean matrices B with all diagonal entries 1 such that B = B^2, which are symmetric about the antidiagonal. These matrices can be seen as closed sets of inversions (pairs (i,j) with 1 <= i < j <= n). A set of inversions E is closed if for all i < j < k, if E contains (i,j) and (j,k) then it contains (i,k).

			The Boolean matrices corresponding to a(4) = 12:
  1 0 0 0    1 0 0 1    1 0 0 0    1 0 0 1
  0 1 0 0    0 1 0 0    0 1 1 0    0 1 1 0
  0 0 1 0    0 0 1 0    0 0 1 0    0 0 1 0
  0 0 0 1    0 0 0 1    0 0 0 1    0 0 0 1
.
  1 0 1 0    1 0 1 1    1 0 1 0    1 0 1 1
  0 1 0 1    0 1 0 1    0 1 1 1    0 1 1 1
  0 0 1 0    0 0 1 0    0 0 1 0    0 0 1 0
  0 0 0 1    0 0 0 1    0 0 0 1    0 0 0 1
.
  1 1 0 0    1 1 0 1    1 1 1 1    1 1 1 1
  0 1 0 0    0 1 0 0    0 1 0 1    0 1 1 1
  0 0 1 1    0 0 1 1    0 0 1 1    0 0 1 1
  0 0 0 1    0 0 0 1    0 0 0 1    0 0 0 1

Cf. A006455, A037223.

def a(n):
    S = set()
    for P in Posets(n):
        if P.is_isomorphic(P.dual()):
            for l in P.linear_extensions():
                t = tuple(tuple(int(P.is_lequal(l[j],l[i])) for j in range(i)) for i in range(1,len(l)))
                if all(t[j][i]==t[n-i-2][n-j-2] for i in range((n-1)//2) for j in range(i,n-i-2)):
                    S.add(t)
    return len(S)

a(10)-a(16) from Christian Sievers, May 02 2025

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A133922 a(n) is number of permutations (p(1),p(2),p(3),...,p(n)) of (1,2,3,...,n) such that p(k) is coprime to p(n+1-k) for k = all positive integers <= n.

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A260189 a(n) = A033148(n) / 2^floor(n/4).

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A277083 Irregular triangle read by rows: T(n,k) = number of size k subsets of S_n that remain unchanged by a rotation of 180 degrees.

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A277085 Irregular triangle read by rows: T(n,k) = number of size k subsets of S_n that remain unchanged by a rotation of 90 degrees.

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A383370 Number of partial orders on {1,2,...,n} that are contained in the usual linear order, whose dual is given by the relabelling k -> n+1-k.

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