A347601 -id:A347601 - cp's OEIS frontend

A346719 a(n) is the number of positive Euler permutations of order 2*n. Bisection (even indices) of A347601.

1, 0, 7, 102, 8109, 642220, 89458803, 15935870034, 3858227881945, 1176448390679256, 447692501190569823, 206713705368363820990, 114132862919751113790597, 74179275137980421348697732, 56081703047542413155379531979, 48790316146471264354636437276330, 48400301382766335524903922737193393

Offset: 0

Peter Luschny, Sep 09 2021

nonn

For definitions and comments see A347601.

Cf. A347601, A346720, A000166, A000364, A347602.

A346719 := n -> (A000166(2*n) + euler(2*n)) / 2:
seq(A346719(n), n = 0..16);

A346719[n_] := Subfactorial[2 n]/2 + Im[PolyLog[-2 n, I]];
Table[A346719[n], {n, 0, 16}]

a(n) + A346720(n) = A000166(2n) (rencontres numbers).
a(n) - A346720(n) = A000364(n) (Euler secant numbers).
a(n) = subfactorial(2*n) / 2 + Im(PolyLog(-2*n, i)).

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).

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

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).

A346720 a(n) is the number of negative Euler permutations of order 2*n. Bisection (even indices) of A347602.

Original entry on oeis.org

0, 1, 2, 163, 6724, 692741, 86756038, 16135231015, 3838836369800, 1178853270354697, 447322130002332298, 206783054242756958891, 114117348385587556703692, 74183362210489714472590093, 56080450787901009525514063694, 48790757690364513377740990959151, 48400123863374755985614486072403728

Offset: 0

Peter Luschny, Sep 09 2021

nonn

For definitions and comments see A347602.

Cf. A347601, A346719, A000166, A000364, A347602.

# uses A000166.
A346720 := n -> (A000166(2*n) - euler(2*n)) / 2:
seq(A346720(n), n = 0..16);

A346720[n_] := Subfactorial[2 n]/2 - Im[PolyLog[-2 n, I]];
Table[A346720[n], {n, 0, 16}]

A346719(n) + a(n) = A000166(2n) (rencontres numbers).
A346719(n) - a(n) = A000364(n) (Euler secant numbers).
a(n) = subfactorial(2*n) / 2 - Im(PolyLog(-2*n, i)).

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

A346719 a(n) is the number of positive Euler permutations of order 2*n. Bisection (even indices) of A347601.

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

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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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A346720 a(n) is the number of negative Euler permutations of order 2*n. Bisection (even indices) of A347602.

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Maple

Mathematica

Formula

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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Maple

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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Author

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Maple