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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A347601 a(n) is the number of positive Euler permutations of order n.

Original entry on oeis.org

1, 0, 0, 2, 7, 16, 102, 1042, 8109, 63280, 642220, 7500626, 89458803, 1135216800, 15935870034, 241410428162, 3858227881945, 65327424977824, 1176448390679256, 22388999178300514, 447692501190569823, 9395318712874789744, 206713705368363820990, 4755693997171333347506
Offset: 0

Views

Author

Peter Luschny, Sep 10 2021

Keywords

Comments

Let M be the tangent matrix of dimension n X n. The definition of the tangent matrix is given in A346831. An Euler permutation of order n is a permutation sigma of {1,...,n} if P = Product_{k=1..n} M(k, sigma(k)) does not vanish. We say sigma is a positive Euler permutation of order n (or sigma in EP(n)) if P = 1 and a negative Euler permutation of order n (or sigma in EN(n)) if P = -1.
a(n) = card(EP(n)), the number of positive Euler permutations of order n. A table of positive Euler permutations is given in A347766. Related sequences are A347599 (Genocchi permutations) and A347600 (Seidel permutations).

Examples

			Illustrating the decomposition of the rencontres numbers and the Euler numbers:
The third column is the sum of the first two columns and the fourth column is the difference between the first two. The fourth column is the sum of the last two.
[n]  A347601     A347602      A000166       A347598      A122045   A347597
--------------------------------------------------------------------------
[ 0] 1,           0,           1,            1,          1,          [0]
[ 1] 0,           0,           0,            0,          0,          0,
[ 2] 0,           1,           1,           -1,         -1,          [0]
[ 3] 2,           0,           2,            2,          0,          2,
[ 4] 7,           2,           9,            5,          5,          [0]
[ 5] 16,          28,          44,          -12,         0,         -12,
[ 6] 102,         163,         265,         -61,        -61,         [0]
[ 7] 1042,        812,         1854,         230,        0,          230,
[ 8] 8109,        6724,        14833,        1385,       1385,       [0]
[ 9] 63280,       70216,       133496,      -6936,       0,         -6936,
[10] 642220,      692741,      1334961,     -50521,     -50521,      [0].

Crossrefs

Cf. A000166, A122045, A346831, A347597, A347598, A347602 (neg. perm.), A347766 (table), A347599, A347600, A346719 (bisection even indices).

Programs

  • Julia
    using Combinatorics
function TangentMatrix(N)
    M = zeros(Int, N, N)
    H = div(N + 1, 2)
    for n in 1:N - 1
        for k in 0:n - 1
            M[n - k, k + 1] = n < H ? 1 : -1
            M[N - n + k + 1, N - k] = n < N - H ? -1 : 1
        end
    end
M end
function EulerPermutations(n, sgn)
    M = TangentMatrix(n)
    S = 0
    for p in permutations(1:n)
        sgn == prod(M[k, p[k]] for k in 1:n) && (S += 1)
    end
S end
PositiveEulerPermutations(n) = EulerPermutations(n, 1)
  • Maple
    # Uses function TangentMatrix from A346831.
EulerPermutations := proc(n, sgn) local M, P, N, s, p, m;
   M := TangentMatrix(n); P := 0; N := 0;
   for p in Iterator:-Permute(n) do
      m := mul(M[k, p(k)], k = 1..n);
      if m =  0 then next fi;
      if m =  1 then P := P + 1 fi;
      if m = -1 then N := N + 1 fi; od;
   if sgn = 'pos' then P else N fi end:
A347601 := n -> `if`(n = 0, 1, EulerPermutations(n, 'pos')):
seq(A347601(n), n = 0..8);

Formula

Let |S| denote the cardinality of a set S. Following identities hold for n >= 0:
a(n) + A347602(n) = |EP(n) | + |EN(n) | = A000166(n) (rencontres numbers),
a(2n) - A347602(2n) = |EP(2n)| - |EN(2n)| = A122045(n) (Euler numbers),
a(n) - A347602(n) = |EP(n) | - |EN(n) | = A347598(n).

A347602 a(n) is the number of negative Euler permutations of order n.

Original entry on oeis.org

0, 0, 1, 0, 2, 28, 163, 812, 6724, 70216, 692741, 7183944, 86756038, 1155576132, 16135231015, 239656087572, 3838836369800, 65522667301840, 1178853270354697, 22361732381344592, 447322130002332298, 9399988542176154796, 206783054242756958891, 4754731473884444589756
Offset: 0

Views

Author

Peter Luschny, Sep 10 2021

Keywords

Comments

Let M be the tangent matrix of dimension n X n. The definition of the tangent matrix is given in A346831. An Euler permutation of order n is a permutation sigma of {1,...,n} if P = Product_{k=1..n} M(k, sigma(k)) does not vanish. We say sigma is a positive Euler permutation of order n (or sigma in EP(n)) if P = 1 and a negative Euler permutation of order n (or sigma in EN(n)) if P = -1.
a(n) = card(EN(n)), the number of negative Euler permutations of order n. A table of negative Euler permutations is given in A347767. Related sequences are A347599 (Genocchi permutations) and A347600 (Seidel permutations).

Crossrefs

Cf. A000166, A122045, A346831, A347597, A347598, A347601 (pos. perm.), A347767 (table), A347599, A347600, A346720 (bisection even indices).

Programs

  • Maple
    # Uses function EulerPermutations from A347601.
A347602 := n -> `if`(n = 0, 0, EulerPermutations(n, 'neg')):
seq(A347602(n), n = 0..8);

Formula

Let |S| denote the cardinality of a set S. Following identities hold for n >= 0:
A347601(n) + a(n) = |EP(n) | + |EN(n) | = A000166(n) (rencontres numbers),
A347601(2n) - a(2n) = |EP(2n)| - |EN(2n)| = A122045(n) (Euler numbers),
A347601(n) - a(n) = |EP(n) | - |EN(n) | = A347598(n).

A347598 a(n) = permanent(T(n)), where T(n) is the tangent matrix defined in A346831 and n >= 1; by convention a(0) = 1.

Original entry on oeis.org

1, 0, -1, 2, 5, -12, -61, 230, 1385, -6936, -50521, 316682, 2702765, -20359332, -199360981, 1754340590, 19391512145, -195242324016, -2404879675441, 27266796955922, 370371188237525, -4669829301365052, -69348874393137901, 962523286888757750, 15514534163557086905
Offset: 0

Views

Author

Peter Luschny, Sep 12 2021

Keywords

Comments

This sequence is an extension of the even-indexed Euler numbers A028296. These numbers can be extended to A000111 by adding the expansion of the tangent function, respectively considering the alternating permutations. Here one gets a different extension of the nonzero Euler numbers by considering the permutations A347601 and A347602 based on the permanent of the tangent matrix as defined in A346831. An overview gives a table in A347601.

Crossrefs

Cf. A346831, A347597, A347601, A347602, A028296, A000111, A122045, A000364.

Programs

  • Maple
    # Uses the function TangentMatrix from A346831.
A347598 := n -> `if`(n = 0, 1, LinearAlgebra:-Permanent(TangentMatrix(n))):
seq(A347598(n), n = 0..12);
  • Sage
    def TangentMatrix(N):
    M = matrix(N, N)
    H = (N + 1) // 2
    for n in range(1, N):
        for k in range(n):
            M[n - k - 1, k] = 1 if n < H else -1
            M[N - n + k, N - k - 1] = -1 if n < N - H else 1
    return M
def A347598(n):
    if n == 0: return 1
    return TangentMatrix(n).permanent()
print([A347598(n) for n in range(12)])

Formula

a(2*n) = A028296(n); a(2*n + 1) = A347597(n).
Showing 1-3 of 3 results.

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