A347600 -id:A347600 - cp's OEIS frontend

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

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

Peter Luschny, Sep 10 2021

nonn

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).

			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].

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

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)

# 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);

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

Peter Luschny, Sep 10 2021

nonn

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).

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

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

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).

A347599 Irregular table read by rows, T(n, k) is the rank of the k-th Genocchi permutation of {1,...,n}, permutations sorted in lexicographical order. If no Genocchi permutation of {1,...,n} exists, then T(n, 1) = 0 by convention.

Original entry on oeis.org

1, 0, 5, 0, 67, 91, 92, 0, 1897, 2017, 2018, 2617, 2619, 2737, 2738, 2739, 2740, 3457, 3458, 3459, 3460, 4177, 4178, 4179, 4180, 0, 99241, 99961, 99962, 104281, 104283, 105001, 105002, 105003, 105004, 110041, 110042, 110043, 110044, 115081, 115082, 115083

Offset: 1

Peter Luschny, Sep 08 2021

Let M be the n X n matrix with M(j, k) = floor((2*j - k ) / n). A Genocchi permutation of order n is a permutation sigma of {1,...,n} if Product_{k=1..n} M(k, sigma(k)) does not vanish.
Let P(n) denote the number of Genocchi permutations of order n. Zhi-Wei Sun conjectured, using permanents, that P(n - 1) = G(n), where G(n) are the Genocchi numbers A036968. From the well-known relation between Genocchi and Bernoulli numbers this implies, assuming the conjecture:
  Bernoulli(n) = P(n - 1) / ((-1)^floor(n/2)*(2^(n + 2) - 2)) for n >= 2.
The related sequence A347600 lists Seidel permutations.

			Table starts:
[1] 1;
[2] 0;
[3] 5;
[4] 0;
[5] 67, 91, 92;
[6] 0;
[7] 1897, 2017, 2018, 2617, 2619, 2737, 2738, 2739, 2740, 3457, 3458, 3459, 3460, 4177, 4178, 4179, 4180;
.
The 17 permutations corresponding to the ranks are for n = 7:
1897 -> [3571246]; 2017 -> [3671245]; 2018 -> [3671254]; 2617 -> [4571236];
2619 -> [4571326]; 2737 -> [4671235]; 2738 -> [4671253]; 2739 -> [4671325];
2740 -> [4671352]; 3457 -> [5671234]; 3458 -> [5671243]; 3459 -> [5671324];
3460 -> [5671342]; 4177 -> [6571234]; 4178 -> [6571243]; 4179 -> [6571324];
4180 -> [6571342].
.
17 / (-510) = -1/30 = Bernoulli(8).

Zhi-Wei Sun, A novel identity connecting permanents to Bernoulli numbers, MathOverflow 2021-09-07.

Cf. A036968, A226158, A027641/A027642, A347600.

using Combinatorics
function GenocchiPermutations(n)
    f(m) = m >= n ? 1 : m < 0 ? -1 : 0
    Mat(n) = [[f(2*j - k) for k in 1:n] for j in 1:n]
    M = Mat(n); P = permutations(1:n); R = Int64[]
    S, rank = 0, 1
    for p in P
        m = prod(M[k][p[k]] for k in 1:n)
        if m != 0
            S += m
            push!(R, rank)
        end
        rank += 1
    end
    # println(n, "  ", S, "  ", S // (2^(n + 2) - 2)) # Bernoulli number
    return R
end
for n in 1:11 println(GenocchiPermutations(n)) end

A347766 Irregular table read by rows, T(n, k) is the rank of the k-th positive Euler permutation of {1,...,n}, permutations sorted in lexicographical order. If no such permutation exists, then T(n, 0) = 0 by convention.

Original entry on oeis.org

1, 0, 0, 2, 3, 1, 6, 8, 11, 14, 15, 17, 3, 8, 24, 28, 29, 30, 32, 35, 50, 55, 57, 68, 71, 74, 79, 92, 2, 6, 15, 16, 21, 26, 30, 40, 44, 54, 55, 60, 68, 99, 104, 120, 121, 123, 124, 125, 137, 138, 142, 143, 144, 146, 150, 161, 164, 167, 174, 175, 177, 179, 185

Offset: 0

Peter Luschny, Sep 12 2021

Let M be the tangent matrix of dimension n X n. The definition of a 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 if P = 1. See A347601 for further details.

A347767 gives the table of negative Euler permutations. Related sequences are A347599 (Genocchi permutations) and A347600 (Seidel permutations).

			Table of positive Euler permutations, length of rows is A347601:
[0] 1;
[1] 0;
[2] 0;
[3] 2, 3;
[4] 1, 6, 8, 11, 14, 15, 17;
[5] 3, 8, 24, 28, 29, 30, 32, 35, 50, 55, 57, 68, 71, 74, 79, 92.
.
The 16 permutations corresponding to the ranks are for n = 5:
    3 -> [12435],  8 -> [13254], 24 -> [15432], 28 -> [21453],
   29 -> [21534], 30 -> [21543], 32 -> [23154], 35 -> [23514],
   50 -> [31254], 55 -> [32145], 57 -> [32415], 68 -> [35142],
   71 -> [35412], 74 -> [41253], 79 -> [42135], 92 -> [45132].

Cf. A347601, A346831, A347767, A347599, A347600.

# Uses function TangentMatrix from A346831.
EulerPermutationsRank := proc(n, sgn) local M, P, N, s, p, m, rank;
   M := TangentMatrix(n); P := []; N := []; rank := 0;
   for p in Iterator:-Permute(n) do
      rank := rank + 1;
      m := mul(M[k, p(k)], k = 1..n);
      if m =  0 then next fi;
      if m =  1 then P := [op(P), rank] fi;
      if m = -1 then N := [op(N), rank] fi; od;
   if sgn = 'pos' then P else N fi end:
A347766Row := n -> `if`(n < 3, [[1,0,0][n+1]], EulerPermutationsRank(n, 'pos')):
for n from 0 to 5 do A347766Row(n) od;

A347767 Irregular table read by rows, T(n, k) is the rank of the k-th negative Euler permutation of {1,...,n}, permutations sorted in lexicographical order. If no such permutation exists, then T(n, 0) = 0 by convention.

Original entry on oeis.org

0, 0, 1, 0, 2, 7, 4, 5, 6, 7, 10, 12, 19, 20, 27, 31, 33, 43, 44, 47, 49, 52, 54, 56, 59, 69, 73, 78, 80, 83, 86, 87, 89, 93, 1, 3, 4, 5, 17, 18, 22, 23, 24, 25, 27, 28, 29, 37, 38, 42, 43, 46, 48, 51, 52, 53, 56, 58, 61, 62, 66, 67, 72, 100, 101, 102, 103, 106

Offset: 0

Peter Luschny, Sep 12 2021

Let M be the tangent matrix of dimension n X n. The definition of a 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 negative Euler permutation of order n if P = -1. See A347601 for further details.

A347766 gives the table of positive Euler permutations. Related sequences are A347599 (Genocchi permutations) and A347600 (Seidel permutations).

			Table of negative Euler permutations, length of rows is A347602:
[0] 0;
[1] 0;
[2] 1;
[3] 0;
[4] 2, 7;
[5] 4, 5, 6, 7, 10, 12, 19, 20, 27, 31, 33, 43, 44, 47, 49, ...
.
The first 8 permutations corresponding to the ranks are for n = 5:
    4 -> [12453],  5 -> [12534],  6 -> [12543],  7 -> [13245],
   10 -> [13452], 12 -> [13542], 19 -> [15234], 20 -> [15243].

Cf. A347601, A347602, A346831, A347766, A347599, A347600.

# Uses function EulerPermutationsRank from A347766.
A347767Row := n -> `if`(n < 4, [[0,0,1,0][n+1]], EulerPermutationsRank(n, 'neg')): for n from 0 to 6 do A347767Row(n) od;

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

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

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A347599 Irregular table read by rows, T(n, k) is the rank of the k-th Genocchi permutation of {1,...,n}, permutations sorted in lexicographical order. If no Genocchi permutation of {1,...,n} exists, then T(n, 1) = 0 by convention.

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A347766 Irregular table read by rows, T(n, k) is the rank of the k-th positive Euler permutation of {1,...,n}, permutations sorted in lexicographical order. If no such permutation exists, then T(n, 0) = 0 by convention.

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A347767 Irregular table read by rows, T(n, k) is the rank of the k-th negative Euler permutation of {1,...,n}, permutations sorted in lexicographical order. If no such permutation exists, then T(n, 0) = 0 by convention.

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