A003112 -id:A003112 - cp's OEIS frontend

A003110 a(n) = number of special odd permutations of 2*n+1.

0, 2, 2, 108, 2028, 32870, 1213110, 46493784, 2310521000, 137466038346, 9842687925450, 832295357128500

Offset: 1

a(n) is the number of odd permutations, pi(0),...,pi(2*n), of 0,...,2*n such that pi(0) = 0 and p(k) = k + pi(k) (mod 2*n+1), k=0,...,2*n is also a permutation of 0,...,2*n. - Sean A. Irvine, Jan 31 2015

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

D. H. Lehmer, Some properties of circulants, J. Number Theory 5 (1973), 43-54.

Cf. A003109 (special even permutations), A003110 (total).

Name clarified and offset corrected by Sean A. Irvine, Jan 31 2015
a(8)-a(9) from Sean A. Irvine, Jan 31 2015
a(10)-a(12) from Vaclav Kotesovec, May 17 2019 (using the terms of A003111 and A003112)

A003109 a(n) = number of special even permutations of 2*n+1.

Original entry on oeis.org

1, 1, 17, 117, 1413, 46389, 1211085, 47977305, 2302999889, 137682614769, 9844042388505, 832087399629125

Offset: 1

N. J. A. Sloane

a(n) is the number of even permutations, pi(0),...,pi(2*n), of 0,...,2*n such that pi(0) = 0 and p(k) = k + pi(k) (mod 2*n+1), k=0,...,2*n is also a permutation of 0,...,2*n. - Sean A. Irvine, Jan 30 2015

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

D. H. Lehmer, Some properties of circulants, J. Number Theory 5 (1973), 43-54.

Cf. A003110 (special odd permutations), A003111 (total).

a(8)-a(9) from Sean A. Irvine, Jan 31 2015

a(10)-a(12) from Vaclav Kotesovec, May 17 2019 (using the terms of A003111 and A003112)

A147679 Triangle read by rows: T(n,k) (n >= 1, 0 <= k <= n-1) is the number of permutations of [0..(n-1)] of spread k.

Original entry on oeis.org

1, 1, 1, 0, 3, 3, 4, 8, 4, 8, 20, 25, 25, 25, 25, 144, 108, 108, 144, 108, 108, 630, 735, 735, 735, 735, 735, 735, 5696, 4608, 5248, 4608, 5696, 4608, 5248, 4608, 39366, 40824, 40824, 39285, 40824, 40824, 39285, 40824, 40824, 366400, 362000, 362000, 362000, 362000, 366400, 362000, 362000, 362000, 362000

Offset: 1

N. J. A. Sloane, May 01 2009

The reference gives more terms, formulas, connection with A003112, etc.

s(pi):= Sum_{j=0..n-1} j*pi(j) (mod j) is defined to be the spread of a permutation pi of [0..(n-1)].

			Triangle begins:
    1
    1   1
    0   3   3
    4   8   4   8
   20  25  25  25  25
  144 108 108 144 108 108
  ...

Seiichi Manyama, Rows n = 1..13, flattened
R. L. Graham and D. H. Lehmer, On the Permanent of Schur's Matrix, J. Australian Math. Soc., 21A (1976), 487-497.

Cf. A003112.
Row sums give: A000142.
Columns k=0-3 give: A004204, A004205, A004206, A004246.
Diagonal gives: A004205.

b:= proc(n) option remember;
     local l, p, r;
     l:= array([i$i=0..n-1]);
     r:= array([0$i=1..n]);
     p:= proc(t,s)
      local d, h, j;
      if t=n then d:= ((s+(n-1)*l[n]) mod n) +1;
                  r[d]:= r[d]+1
      else for j from t to n do
            l[t],l[j]:= l[j],l[t];
            p(t+1, (s+(t-1)*l[t]) )
           od;
           h:= l[t];
           for j from t to n-1 do l[j]:= l[j+1] od;
           l[n]:= h
      fi
     end;
     p(1,0);
     eval(r)
    end:
T:= (n,k)-> b(n)[k+1]:
seq (seq (T(n,k), k=0..n-1), n=1..10);

b[n_] := b[n] = Module[{l, p, r}, l = Range[0, n-1]; r = Array[0&, n]; p [t_, s_] := Module[{d, h, j}, If[t == n, d = Mod[s+(n-1)*l[[n]], n]+1; r[[d]] = r[[d]]+1, For[j = t, j <= n, j++, {l[[t]], l[[j]]} = {l[[j]], l[[t]]}; p[t+1, s+(t-1)*l[[t]]]]; h = l[[t]]; For[j = t, j <= n-1, j++, l[[j]] = l[[j+1]]]; l[[n]] = h]]; p[1, 0]; r]; t[n_, k_] := b[n][[k+1]]; Table [Print[t[n, k]]; t[n, k], {n, 1, 10}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, Apr 17 2014, after Alois P. Heinz *)

@CachedFunction
def A147679_row(n):
    row = [0]*n
    for p in Permutations(range(n)):
        spread = sum(i*px for i,px in enumerate(p)) % n
        row[spread] += 1
    return row
A147679 = lambda n,k: A147679_row(n)[k] # D. S. McNeil, Dec 23 2010

Edited by Alois P. Heinz, Dec 22 2010

A346949 Value of the permanent of the matrix [1-zeta^{j-k}]_{1<=j,k<=2n}, where zeta is any primitive 2n-th root of unity.

Original entry on oeis.org

4, 48, 1440, 80640, 7257600, 958003200, 174356582400, 41845579776000, 12804747411456000, 4865804016353280000, 2248001455555215360000, 1240896803466478878720000, 806582922253211271168000000, 609776689223427721003008000000, 530505719624382117272616960000000, 526261673867387060334436024320000000

Offset: 1

Zhi-Wei Sun, Aug 08 2021

nonn

The author has proved that the exact value of a(n) is 2*(2n)!. Moreover, for any primitive n-th root zeta of unity, the permanent of the matrix [1-zeta^j*x_k]_{1<=j,k<=n} is n!(1-x_1..x_n).

Conjecture: Let zeta be a primitive 2n-th root of unity. Then the sum of those Product_{j=1..2n}(1-zeta^{j-f(j)})^{-1} with f over all the derangements of {1,...,2n} has the exact value ((2n-1)!!/2^n)^2.

			a(1) is the permanent of the matrix [1-(-1)^{1-1},1-(-1)^{1-2};1-(-1)^{2-1},1-(-1)^{2-2}] = [0,2;2,0], which equals 4.

Zhi-Wei Sun, Arithmetic properties of some permanents, arXiv:2108.07723 [math.GM], 2021.

Cf. A000166, A003112, A204249, A278847, A346934.

a[n_]:=a[n]= Permanent[Table[1-E^(2*Pi*I*(j-k)/(2*n)),{j,1,2n},{k,1,2n}]];
(* Though a(n) is actually an integer, Mathematica could not find its exact value for a general positive integer n. Instead, we may check approximate values of a(n) such as N[a[5],10] = 7257600.000. *)

default(realprecision, 100); a(n) = round(real(matpermanent(matrix(2*n, 2*n, j, k, 1-exp(Pi*I*(j-k)/n))))) \\ Michel Marcus, Aug 08 2021

a(n) = 2*(2*n)!.

a(16) from Vaclav Kotesovec, Aug 21 2021

cp's OEIS Frontend

A003110 a(n) = number of special odd permutations of 2*n+1.

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A003109 a(n) = number of special even permutations of 2*n+1.

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A147679 Triangle read by rows: T(n,k) (n >= 1, 0 <= k <= n-1) is the number of permutations of [0..(n-1)] of spread k.

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A346949 Value of the permanent of the matrix [1-zeta^{j-k}]_{1<=j,k<=2n}, where zeta is any primitive 2n-th root of unity.

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Programs

Mathematica

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Extensions