A346831 -id:A346831 - cp's OEIS frontend

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

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

Peter Luschny, Sep 12 2021

sign

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.

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

# Uses the function TangentMatrix from A346831.
A347598 := n -> `if`(n = 0, 1, LinearAlgebra:-Permanent(TangentMatrix(n))):
seq(A347598(n), n = 0..12);

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)])

a(2*n) = A028296(n); a(2*n + 1) = A347597(n).

A347597 a(n) = Permanent(T(2*n + 1)) where T(n) is the tangent matrix defined in A346831. Bisection of A347598 (odd indices).

Original entry on oeis.org

0, 2, -12, 230, -6936, 316682, -20359332, 1754340590, -195242324016, 27266796955922, -4669829301365052, 962523286888757750, -235056895264868039496, 67119429827860669710362

Offset: 0

Peter Luschny, Sep 14 2021

Cf. A347598, A346831.

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

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

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).

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).

A346837 Table read by rows, coefficients of the determinant polynomials of the generalized tangent matrices.

Original entry on oeis.org

1, 0, 1, -1, 0, -1, -2, -1, 0, -1, 1, 0, 6, 0, 1, -4, 1, 12, 6, 0, 1, -1, 0, -15, 0, -15, 0, -1, -14, -17, 12, 1, -30, -15, 0, -1, 1, 0, 28, 0, 70, 0, 28, 0, 1, -40, -63, 72, 156, 40, 6, 56, 28, 0, 1

Offset: 0

Peter Luschny, Sep 11 2021

The generalized tangent matrix M(n, k) is an N X N matrix defined for n in [1..N-1] and for k in [0..n-1] with h = floor((N+1)/2) as:
   M[n - k, k + 1] = if n < h then 1 otherwise -1,
   M[N - n + k + 1, N - k] = if n < N - h then -1 otherwise 1,
and the indeterminate x in the main antidiagonal.
The tangent matrix M(n, k) as defined in A346831 is the special case which arises from setting x = 0. The determinant of a generalized tangent matrix M is a polynomial which we call the determinant polynomial of M.

			Table starts:
[0]   1;
[1]   0,   1;
[2]  -1,   0,  -1;
[3]  -2,  -1,   0,  -1;
[4]   1,   0,   6,   0,   1;
[5]  -4,   1,  12,   6,   0,   1;
[6]  -1,   0, -15,   0, -15,   0, -1;
[7] -14, -17,  12,   1, -30, -15,  0, -1;
[8]   1,   0,  28,   0,  70,   0, 28,  0, 1;
[9] -40, -63,  72, 156,  40,   6, 56, 28, 0, 1.
.
The first few generalized tangent matrices:
1       2          3              4                  5
---------------------------------------------------------------
x;   -1  x;    1  -1  x;    1  -1  -1   x;   1   1  -1  -1   x;
      x  1;   -1   x  1;   -1  -1   x   1;   1  -1  -1   x   1;
               x   1  1;   -1   x   1   1;  -1  -1   x   1   1;
                            x   1   1  -1;  -1   x   1   1   1;
                                             x   1   1   1  -1;

Cf. A011782 (row sums modulo sign), A347596 (alternating row sums), A346831.

GeneralizedTangentMatrix := proc(N) local M, H, n, k;
   M := Matrix(N, N); H := iquo(N + 1, 2);
   for n from 1 to N - 1 do for k from 0 to n - 1 do
       M[n - k, k + 1] := `if`(n < H, 1, -1);
       M[N - n + k + 1, N - k] := `if`(n < N - H, -1, 1);
od od; for k from 1 to N do M[k, N-k+1] := x od;
M end:
A346837Row := proc(n) if n = 0 then return 1 fi;
   GeneralizedTangentMatrix(n):
   LinearAlgebra:-Determinant(%);
   seq(coeff(%, x, k), k = 0..n) end:
seq(A346837Row(n), n = 0..9);

GeneralizedTangentMatrix[N_] := Module[{M, H, n, k},
   M = Array[0&, {N, N}]; H = Quotient[N + 1, 2];
   For[n = 1, n <= N - 1, n++, For[k = 0, k <= n - 1, k++,
      M[[n - k, k + 1]] = If[n < H, 1, -1];
      M[[N - n + k + 1, N - k]] = If[n < N - H, -1, 1]]];
      For[k = 1, k <= N, k++, M[[k, N - k + 1]] = x]; M];
A346837Row[n_] := If[n == 0, {1}, CoefficientList[ Det[
   GeneralizedTangentMatrix[n]], x]];
Table[A346837Row[n], {n, 0, 9}] // Flatten (* Jean-François Alcover, Apr 15 2024, after Peter Luschny *)

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;

cp's OEIS Frontend

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

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A347597 a(n) = Permanent(T(2*n + 1)) where T(n) is the tangent matrix defined in A346831. Bisection of A347598 (odd indices).

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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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A346837 Table read by rows, coefficients of the determinant polynomials of the generalized tangent matrices.

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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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