A085807 -id:A085807 - cp's OEIS frontend

A353453 a(n) is the permanent of the n X n symmetric matrix M(n) that is defined as M[i,j] = abs(i - j) if min(i, j) < max(i, j) <= 2*min(i, j), and otherwise 0.

1, 0, 1, 0, 1, 4, 64, 576, 7844, 63524, 882772, 11713408, 252996564, 5879980400, 184839020672, 5698866739200, 229815005974352, 9350598794677712, 480306381374466176, 23741710999960266176, 1446802666239931811472, 86153125248221968292928, 6197781268948296566634304

Offset: 0

Stefano Spezia, Apr 19 2022

nonn

			a(8) = 7844:
    0,  1,  0,  0,  0,  0,  0,  0;
    1,  0,  1,  2,  0,  0,  0,  0;
    0,  1,  0,  1,  2,  3,  0,  0;
    0,  2,  1,  0,  1,  2,  3,  4;
    0,  0,  2,  1,  0,  1,  2,  3;
    0,  0,  3,  2,  1,  0,  1,  2;
    0,  0,  0,  3,  2,  1,  0,  1;
    0,  0,  0,  4,  3,  2,  1,  0.

Cf. A085750, A085807.

Cf. A000982 (number of zero matrix elements), A003983, A006918, A007590 (number of positive matrix elements), A049581, A051125, A173997, A350050, A352967, A353452 (determinant).

Join[{1},Table[Permanent[Table[If[Min[i,j]

a(n) = matpermanent(matrix(n, n, i, j, if ((min(i,j) < max(i,j)) && (max(i,j) <= 2*min(i,j)), abs(i-j)))); \\ Michel Marcus, Apr 20 2022

from sympy import Matrix
def A353453(n): return Matrix(n, n, lambda i, j: abs(i-j) if min(i,j)Chai Wah Wu, Aug 29 2023

Sum_{i=1..n+1-k} M[i,i+k] = A173997(n, k) with 1 <= k <= floor((n + 1)/2).
Sum_{i=1..n} Sum_{j=1..n} M[i,j] = 2*A006918(n-1).
Sum_{i=1..n} Sum_{j=1..n} M[i,j]^2 = A350050(n+1).

A374283 a(n) is the maximal permanent of an n X n symmetric Toeplitz matrix having 0 on the main diagonal and all the integers 1, 2, ..., n-1 off-diagonal.

Original entry on oeis.org

1, 0, 1, 8, 256, 9978, 600052, 49036950, 5286564352, 725724599636

Offset: 0

Stefano Spezia, Jul 02 2024

			a(5) = 9978:
  [0, 4, 3, 2, 1]
  [4, 0, 4, 3, 2]
  [3, 4, 0, 4, 3]
  [2, 3, 4, 0, 4]
  [1, 2, 3, 4, 0]

Wikipedia, Toeplitz Matrix.

Cf. A085807 (minimal), A358327.

Cf. A374279, A374280, A374281, A374282.

a[0]=1; a[n_]:=Max[Table[Permanent[ToeplitzMatrix[Join[{0}, Part[Permutations[Range[n - 1]], i]]]], {i, (n-1)!}]]; Array[a, 11, 0]

A374140 a(n) is the permanent of the symmetric Toeplitz matrix of order n whose element (i,j) equals abs(i-j) or 1 if i = j.

Original entry on oeis.org

1, 1, 2, 11, 117, 2083, 55482, 2063149, 102176977, 6490667261, 514651043730, 49787897503031, 5771746960693493, 789652404867861919, 125885777192807718730, 23129357587464094132601, 4851600400570400272371009, 1152232847579194480216644249, 307579355879152834353840187554

Offset: 0

Stefano Spezia, Jun 28 2024

nonn

Conjecture: a(n) is the minimal permanent of an n X n symmetric Toeplitz matrix having 1 on the main diagonal and all the integers 1, 2, ..., n-1 off-diagonal. - Stefano Spezia, Jul 05 2024

			a(4) = 117:
  [1, 1, 2, 3]
  [1, 1, 1, 2]
  [2, 1, 1, 1]
  [3, 2, 1, 1]

Cf. A085807, A374067, A374139 (determinant).

a[n_]:=Permanent[Table[If[i == j, 1, Abs[i - j]], {i, n}, {j, n}]]; Join[{1}, Array[a, 18]]

a(n) = matpermanent(matrix(n, n, i, j, if (i==j, 1, abs(i-j)))); \\ Michel Marcus, Jun 29 2024

from sympy import Matrix
def A374140(n): return Matrix(n,n,[abs(j-k) if j!=k else 1 for j in range(n) for k in range(n)]).per() if n else 1 # Chai Wah Wu, Jul 01 2024

A278857 a(n) = permanent M_n where M_n is the n X n matrix m(i,j) = (i-j)^2.

Original entry on oeis.org

1, 0, 1, 8, 676, 49600, 10335908, 2658757248, 1214367336000, 730771063280640, 642638269862752320, 736176718456263406080, 1122592471007868379259136, 2168016139899273930219233280, 5288852927890824307509101287680, 15889369670472598370104100032512000

Offset: 0

Vaclav Kotesovec, Nov 29 2016

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..37

Cf. A085750, A085807, A204249, A278845, A278847, A278858.

with(LinearAlgebra):
a:= n-> `if`(n=0, 1, Permanent(Matrix(n, (i, j)-> (i-j)^2))):
seq(a(n), n=0..16);  # Vaclav Kotesovec, Nov 30 2016, after Alois P. Heinz

Flatten[{1, Table[Permanent[Table[(i-j)^2, {i, 1, n}, {j, 1, n}]], {n, 1, 15}]}]

{a(n) = matpermanent(matrix(n, n, i, j, (i-j)^2))}
for(n=0, 20, print1(a(n), ", ")) \\ Vaclav Kotesovec, Aug 12 2021

A278858 a(n) = permanent M_n where M_n is the n X n matrix m(i,j) = abs(i^2-j^2).

Original entry on oeis.org

1, 0, 9, 240, 36864, 7741440, 3363235524, 2203143038208, 2248347011420160, 3260265586467690240, 6578570637254005920000, 17755898734939822501524480, 62673017366111480630785474560, 282641923592380319367599892725760, 1599753679036773033206787507696238848

Offset: 0

Vaclav Kotesovec, Nov 29 2016

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..36

Cf. A085750, A085807, A204249, A278845, A278847, A278857.

Flatten[{1, Table[Permanent[Table[Abs[i^2-j^2], {i, 1, n}, {j, 1, n}]], {n, 1, 15}]}]

{a(n) = matpermanent(matrix(n, n, i, j, abs(i^2-j^2)))}
for(n=0, 20, print1(a(n), ", ")) \\ Vaclav Kotesovec, Aug 12 2021

A347768 Permanent of the n X n matrix with (j,k)-entry |j-k+1| (j,k = 1..n).

Original entry on oeis.org

1, 1, 8, 80, 1568, 42308, 1632544, 82377984, 5340729728, 429905599744, 42164608801024, 4944386388782080, 683353973472423936, 109907353260811403520, 20352830852731108528128, 4299139435513999926820864, 1027450150728092835655335936, 275824741022588671077713641472

Offset: 1

Zhi-Wei Sun, Sep 12 2021

nonn

Conjecture: For any odd prime p, we have a(p) == 1/2 (mod p).

It is easy to show that for any integer n > 1 the determinant of the n X n matrix with (j,k)-entry |j-k+1| (j,k=1..n) has the value 2^(n-2).

			a(2) = 1 since the permanent of the matrix [|1-1+1|,|1-2+1|; |2-1+1|,|2-2+1|] = [1,0;2,1] has the value 1.

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

Cf. A085807.

a[n_]:=a[n]=Permanent[Table[Abs[j-k+1],{j,1,n},{k,1,n}]]
Table[a[n],{n,1,22}]

a(n) = matpermanent(matrix(n, n, j, k, abs(j-k+1))); \\ Michel Marcus, Sep 13 2021

from sympy import Matrix
def A347768(n): return Matrix(n,n,[abs(j-k+1) for j in range(n) for k in range(n)]).per() # Chai Wah Wu, Sep 14 2021

A357503 a(n) is the hafnian of the 2n X 2n symmetric matrix whose element (i,j) equals abs(i-j).

Original entry on oeis.org

1, 1, 8, 174, 7360, 512720, 53245824, 7713320944, 1486382446592, 367691598791424, 113570289012090880

Offset: 0

Stefano Spezia, Oct 01 2022

			a(2) = M_{1,2}*M_{3,4} + M_{1,3}*M_{2,4} + M_{1,4}*M_{2,3} = 8 is the hafnian of
    0, 1, 2, 3;
    1, 0, 1, 2;
    2, 1, 0, 1;
    3, 2, 1, 0.

Wikipedia, Hafnian
Wikipedia, Symmetric matrix

Cf. A049581, A085750 (determinant of M(n)), A085807 (permanent of M(n)), A094053 (super- and subdiagonal sums of M(n) in reversed order), A144216 (row- and column sums of M(n)), A338456.

M[i_, j_, n_]:=Part[Part[Table[Abs[r-c], {r, n}, {c, n}], i], j]; a[n_]:=Sum[Product[M[Part[PermutationList[s, 2n], 2i-1], Part[PermutationList[s, 2n], 2i], 2n], {i, n}], {s, SymmetricGroup[2n]//GroupElements}]/(n!*2^n); Array[a, 6, 0]

tm(n) = matrix(n, n, i, j, abs(i-j));
a(n) = my(m = tm(2*n), s=0); forperm([1..2*n], p, s += prod(j=1, n, m[p[2*j-1], p[2*j]]); ); s/(n!*2^n); \\ Michel Marcus, May 02 2023

a(6) from Michel Marcus, May 02 2023

a(7)-a(10) from Pontus von Brömssen, Oct 15 2023

A381514 a(n) is the hafnian of a symmetric Toeplitz matrix of order 2*n whose off-diagonal element (i,j) equals the |i-j|-th prime.

Original entry on oeis.org

1, 2, 23, 899, 85072, 15120411, 4439935299, 1989537541918, 1264044973158281, 1090056235155152713, 1227540523199054294506

Offset: 0

Stefano Spezia, Feb 25 2025

			a(2) = 23 because the hafnian of
  [d  2  3  5]
  [2  d  2  3]
  [3  2  d  2]
  [5  3  2  d]
equals M_{1,2}*M_{3,4} + M_{1,3}*M_{2,4} + M_{1,4}*M_{2,3} = 2*2 + 3*3 + 5*2 = 23. Here d denotes the generic element on the main diagonal of the matrix from which the hafnian does not depend.

Wikipedia, Hafnian.
Wikipedia, Symmetric matrix.
Wikipedia, Toeplitz Matrix.

Cf. A374067, A374068.

Cf. A071078, A071079, A085807, A306457, A318173, A356483.

M[i_, j_]:=Prime[Abs[i-j]]; a[n_]:=Sum[Product[M[Part[PermutationList[s, 2n], 2i-1], Part[PermutationList[s, 2n], 2i]], {i, n}], {s, SymmetricGroup[2n]//GroupElements}]/(n!*2^n); Array[a, 5, 0]

a(5)-a(10) from Pontus von Brömssen, Feb 26 2025

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A353453 a(n) is the permanent of the n X n symmetric matrix M(n) that is defined as M[i,j] = abs(i - j) if min(i, j) < max(i, j) <= 2*min(i, j), and otherwise 0.

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A374283 a(n) is the maximal permanent of an n X n symmetric Toeplitz matrix having 0 on the main diagonal and all the integers 1, 2, ..., n-1 off-diagonal.

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A374140 a(n) is the permanent of the symmetric Toeplitz matrix of order n whose element (i,j) equals abs(i-j) or 1 if i = j.

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A278857 a(n) = permanent M_n where M_n is the n X n matrix m(i,j) = (i-j)^2.

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A278858 a(n) = permanent M_n where M_n is the n X n matrix m(i,j) = abs(i^2-j^2).

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A347768 Permanent of the n X n matrix with (j,k)-entry |j-k+1| (j,k = 1..n).

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A357503 a(n) is the hafnian of the 2n X 2n symmetric matrix whose element (i,j) equals abs(i-j).

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A381514 a(n) is the hafnian of a symmetric Toeplitz matrix of order 2*n whose off-diagonal element (i,j) equals the |i-j|-th prime.

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