A204249 -id:A204249 - cp's OEIS frontend

A278840 a(n) = permanent M_n where M_n is the n X n matrix m(i,j) = A000041(i+j).

1, 2, 19, 642, 58884, 13569779, 6931351962, 7532494931779, 16299546505518855, 67814300022651169814, 520884812091898994319805, 7206655416715261673779120809, 174009567319884878178189603283634, 7196671016523025599652036668556922867

Offset: 0

Vaclav Kotesovec, Nov 29 2016

nonn

Cf. A000041, A005249, A204249, A278838, A278841.

Flatten[{1, Table[Permanent[Table[PartitionsP[i+j], {i, 1, n}, {j, 1, n}]], {n, 1, 14}]}]

A278841 a(n) = permanent M_n where M_n is the n X n matrix m(i,j) = A000009(i+j).

Original entry on oeis.org

1, 1, 6, 65, 1737, 91359, 8755730, 1465091787, 420070484342, 194857695312573, 142349343815684947, 161388097061567486595, 276760372468557882285737, 707850058213409589011565269, 2654427644322345709705054800083

Offset: 0

Vaclav Kotesovec, Nov 29 2016

nonn

Cf. A000009, A005249, A204249, A278839, A278840.

Flatten[{1, Table[Permanent[Table[PartitionsQ[i+j], {i, 1, n}, {j, 1, n}]], {n, 1, 14}]}]

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

A346949 Value of the permanent of the matrix [1-zeta^{j-k}]_{1<=j,k<=2n}, where zeta is any primitive 2n-th root of unity.

Original entry on oeis.org

4, 48, 1440, 80640, 7257600, 958003200, 174356582400, 41845579776000, 12804747411456000, 4865804016353280000, 2248001455555215360000, 1240896803466478878720000, 806582922253211271168000000, 609776689223427721003008000000, 530505719624382117272616960000000, 526261673867387060334436024320000000

Offset: 1

Zhi-Wei Sun, Aug 08 2021

nonn

The author has proved that the exact value of a(n) is 2*(2n)!. Moreover, for any primitive n-th root zeta of unity, the permanent of the matrix [1-zeta^j*x_k]_{1<=j,k<=n} is n!(1-x_1..x_n).

Conjecture: Let zeta be a primitive 2n-th root of unity. Then the sum of those Product_{j=1..2n}(1-zeta^{j-f(j)})^{-1} with f over all the derangements of {1,...,2n} has the exact value ((2n-1)!!/2^n)^2.

			a(1) is the permanent of the matrix [1-(-1)^{1-1},1-(-1)^{1-2};1-(-1)^{2-1},1-(-1)^{2-2}] = [0,2;2,0], which equals 4.

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

Cf. A000166, A003112, A204249, A278847, A346934.

a[n_]:=a[n]= Permanent[Table[1-E^(2*Pi*I*(j-k)/(2*n)),{j,1,2n},{k,1,2n}]];
(* Though a(n) is actually an integer, Mathematica could not find its exact value for a general positive integer n. Instead, we may check approximate values of a(n) such as N[a[5],10] = 7257600.000. *)

default(realprecision, 100); a(n) = round(real(matpermanent(matrix(2*n, 2*n, j, k, 1-exp(Pi*I*(j-k)/n))))) \\ Michel Marcus, Aug 08 2021

a(n) = 2*(2*n)!.

a(16) from Vaclav Kotesovec, Aug 21 2021

A278927 a(n) = permanent M_n where M_n is the n X n matrix m(i,j) = 2*i + j.

Original entry on oeis.org

1, 3, 38, 1116, 59392, 5004720, 613252320, 103050420480, 22752244279296, 6388491978086400, 2223423557203968000, 939489529945565491200, 473789563269835667374080, 281112352557447776249364480, 193857685859605294233907200000, 153758529080702011472247521280000

Offset: 0

Vaclav Kotesovec, Dec 01 2016

nonn

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

Cf. A204249.

with(LinearAlgebra):
a:= n-> `if`(n=0, 1, Permanent(Matrix(n, (i, j)-> 2*i+j))):
seq(a(n), n=0..16);

Flatten[{1, Table[Permanent[Table[2*i+j, {i, 1, n}, {j, 1, n}]], {n, 1, 15}]}]

{a(n) = matpermanent(matrix(n, n, i, j, 2*i+j))}
for(n=0, 20, print1(a(n), ", ")) \\ Vaclav Kotesovec, Dec 21 2018

a(n) ~ c * d^n * n!^2 / sqrt(n), where d = 3.63208011334048289... and c = 1.47836065972078...

A356041 Numerator of the permanent of the matrix [m(j,k)]_{j,k=0..2*n}, where m(j,k) is 1 or (j+k)/(j-k) according as j = k or not.

Original entry on oeis.org

1, -10, 5870, -436619903, 204409938157631, -445211270783816444430983, 396293055917701185511708646165266818589, -50075241686736491417389691481681057226117081701278447, 5775845962974711619751142332219720943292883679628990682130802753216873

Offset: 0

Zhi-Wei Sun, Jul 24 2022

sign

Conjecture 1. If n is a positive odd integer, x(0) = 0, and x(1),...,x(n) are variables, then the permanent of the matrix [x(j,k)]_{j,k=0..n} vanishes, where x(j,k) is 1 or (x(j)+x(k))/(x(j)-x(k)) according as j = k or not.
Conjecture 2: (-1)^n*a(n) > 0 for all n = 0,1,2,...
Conjecture 3: For any odd prime p, the permanent of the matrix [m(j,k)]_{j,k=0..p-1} is congruent to 1 - (-1)^((p-1)/2) modulo p, where m(j,k) is defined as in the title..
Conjecture 4: For any odd prime p, the permanent of the matrix M = [m(j,k)]_{j,k=1..p-1} is congruent to ((p-2)!!)^2 modulo p^2, and det(M) is congruent to ((p-2)!!)^2*(-1)^((p+1)/2)/(p-2) modulo p^2.

			a(1) = -10 since the permanent of the matrix [m(j,k)]_{j,k=0,1,2} = [1,-1,1; 1,1,-3; 1,3,1] is -10.

Yue-Feng She, Zhi-Wei Sun, and Wei Xia, A novel permanent identity, arXiv:2208.12167 [math.CO], 2022.
Zhi-Wei Sun, New Conjectures in Number Theory and Combinatorics (in Chinese), Harbin Institute of Technology Press, 2021.
Zhi-Wei Sun, Arithmetic properties of some permanents, arXiv:2108.07723 [math.GM], 2021.

Cf. A204249.

a[n_]:=a[n]=Numerator[Permanent[Table[If[j==k,1,(j+k)/(j-k)],{j,0,2n},{k,0,2n}]]]
Table[a[n],{n,0,8}]

a(n) = numerator(matpermanent(matrix(2*n+1, 2*n+1, i, j, i--; j--; if (i==j, 1, (i+j)/(i-j))))); \\ Michel Marcus, Jul 24 2022

cp's OEIS Frontend

A278840 a(n) = permanent M_n where M_n is the n X n matrix m(i,j) = A000041(i+j).

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A278841 a(n) = permanent M_n where M_n is the n X n matrix m(i,j) = A000009(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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A346949 Value of the permanent of the matrix [1-zeta^{j-k}]_{1<=j,k<=2n}, where zeta is any primitive 2n-th root of unity.

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A278927 a(n) = permanent M_n where M_n is the n X n matrix m(i,j) = 2*i + j.

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A356041 Numerator of the permanent of the matrix [m(j,k)]_{j,k=0..2*n}, where m(j,k) is 1 or (j+k)/(j-k) according as j = k or not.

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