A383772
a(n) = neg(M(n)), where M(n) is the n X n circulant matrix with (row 1) = (1, 2, ... , n), and neg(M(n)) is the negative part of the determinant of M(n); see A380661.
Original entry on oeis.org
0, -4, -18, -610, -15675, -772122, -47282844, -3918873376, -410168886615, -53329052728000, -8417451284317614, -1586200451151892608, -351735180091505203539, -90667510133054591492224, -26884188746929397888775000, -9086147134545912835276742656
Offset: 1
The rows of M(4) are (1, 2, 3, 4), (4, 1, 2, 3), (3, 4, 1, 2), (2, 3, 4, 1); determinant(M(4)) = -160; permanent(M(4)) = 1060, so neg(M(4)) = (-160 - 1060)/2 = -610 and pos(M(4)) = (-160 + 1060)/2 = 450.
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z = 18;
v[n_] := Table[k + 1, {k, 0, n - 1}];
u[n_] := Table[RotateRight[#, k - 1], {k, 1, Length[#]}] &[v[n]];
p = Table[Simplify[Permanent[u[n]]], {n, 1, z}] (* A085719 *)
d = Table[Simplify[Det[u[n]]], {n, 1, z}] (* A052182 *)
neg = (d - p)/2 (* A383772 *)
pos = (d + p)/2 (* A383773 *)
A383773
a(n) = pos(M(n)), where M(n) is the n X n circulant matrix with (row 1) = (1, 2, ... , n), and pos(M(n)) is the positive part of the determinant of M(n); see A380661.
Original entry on oeis.org
1, 1, 36, 450, 17550, 744906, 47753440, 3909436192, 410384120220, 53323552728000, 8417606908865220, 1586195621597483136, 351735343178101060906, 90667504180193792086144, 26884188980472806091900000, 9086147124746080046118543360, 3472279409772212369077001352888
Offset: 1
The rows of M(4) are (1, 2, 3, 4), (4, 1, 2, 3), (3, 4, 1, 2), (2, 3, 4, 1); determinant(M(4)) = -160; permanent(M(4)) = 1060, so neg(M(4)) = (-160 - 1060)/2 = -610 and pos(M(4)) = (-160 + 1060)/2 = 450.
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z = 18;
v[n_] := Table[k + 1, {k, 0, n - 1}];
u[n_] := Table[RotateRight[#, k - 1], {k, 1, Length[#]}] &[v[n]];
p = Table[Simplify[Permanent[u[n]]], {n, 1, z}] (* A085719 *)
d = Table[Simplify[Det[u[n]]], {n, 1, z}] (* A052182 *)
neg = (d - p)/2 (* A383772 *)
pos = (d + p)/2 (* A383773 *)
A383774
a(n) = neg(M(n)), where M(n) is the n X n left circulant matrix with (row 1) = (1, 2, ... , n), and neg(M(n)) is the negative part of the determinant of M(n); see A380661.
Original entry on oeis.org
0, -4, -36, -450, -15675, -772122, -47753440, -3909436192, -410168886615, -53329052728000, -8417606908865220, -1586195621597483136, -351735180091505203539, -90667510133054591492224, -26884188980472806091900000, -9086147124746080046118543360
Offset: 1
The rows of M(4) are (1, 2, 3, 4), (2, 3, 4, 1), (3, 4, 1, 2), (4, 1, 2, 3); determinant(M(4)) = 160; permanent(M(4)) = 1060, so neg(M(4)) = (160 - 1060)/2 = -450 and pos(M(4)) = (160 + 1060)/2 = 610.
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z = 18;
v[n_] := Table[k + 1, {k, 0, n - 1}];
u[n_] := Table[RotateLeft[#, k - 1], {k, 1, Length[#]}] &[v[n]];
p = Table[Simplify[Permanent[u[n]]], {n, 1, z}] (* A085719 *)
d = Table[Simplify[Det[u[n]]], {n, 1, z}] (* A052182, with altered signs *)
neg = (d - p)/2 (* A383774 *)
pos = (d + p)/2 (* A383775 *)
A383775
a(n) = pos(M(n)), where M(n) is the n X n left circulant matrix with (row 1) = (1, 2, ... , n), and pos(M(n)) is the positive part of the determinant of M(n); see A380661.
Original entry on oeis.org
1, 1, 18, 610, 17550, 744906, 47282844, 3918873376, 410384120220, 53323552728000, 8417451284317614, 1586200451151892608, 351735343178101060906, 90667504180193792086144, 26884188746929397888775000, 9086147134545912835276742656, 3472279409772212369077001352888
Offset: 1
The rows of M(4) are (1, 2, 3, 4), (2, 3, 4, 1), (3, 4, 1, 2), (4, 1, 2, 3); determinant(M(4)) = 160; permanent(M(4)) = 1060, so neg(M(4)) = (160 - 1060)/2 = -450 and pos(M(4)) = (160 + 1060)/2 = 610.
-
z = 18;
v[n_] := Table[k + 1, {k, 0, n - 1}];
u[n_] := Table[RotateLeft[#, k - 1], {k, 1, Length[#]}] &[v[n]];
p = Table[Simplify[Permanent[u[n]]], {n, 1, z}] (* A085719 *)
d = Table[Simplify[Det[u[n]]], {n, 1, z}] (* A052182, with altered signs *)
neg = (d - p)/2 (* A383774 *)
pos = (d + p)/2 (* A383775 *)
A086759
Permanent of the Cayley addition table of Z_{n}. a(n) is the permanent of the n X n matrix M_(i,j) = ((i+j) mod n) where i and j range from 0 to n-1.
Original entry on oeis.org
0, 1, 9, 164, 5050, 227508, 14064519, 1146668608, 119249333028, 15400125776000, 2417814003691405, 453536611741073664, 100178077459552487070, 25735749696251388478720, 7608415981499790110521875, 2564724413131659780025106432, 977834710569917222742633274504
Offset: 1
Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 01 2003
a(9) is the permanent of the matrix
0 1 2 3 4 5 6 7 8
1 2 3 4 5 6 7 8 0
2 3 4 5 6 7 8 0 1
3 4 5 6 7 8 0 1 2
4 5 6 7 8 0 1 2 3
5 6 7 8 0 1 2 3 4
6 7 8 0 1 2 3 4 5
7 8 0 1 2 3 4 5 6
8 0 1 2 3 4 5 6 7
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Array[With[{s = Range[0, #]}, Permanent@ Array[RotateLeft[s, #] &, Last@ s + 1, 0]] &, 16, 0] (* Michael De Vlieger, Sep 03 2019 *)
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permRWNb(a)=n=matsize(a)[1];if(n==1,return(a[1,1]));sg=1;in=vectorv(n);x=in;x=a[,n]-sum(j=1,n,a[,j])/2;p=prod(i=1,n,x[i]);for(k=1,2^(n-1)-1,sg=-sg;j=valuation(k,2)+1;z=1-2*in[j];in[j]+=z;x+=z*a[,j];p+=prod(i=1,n,x[i],sg));return(2*(2*(n%2)-1)*p) for(n=1,21,a=matrix(n,n,i,j,((i+j-2)%n));print1(permRWNb(a)",")) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), May 14 2007
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a(n) = matpermanent(matrix(n, n, i, j, (i+j-2) % n)) \\ Stefano Spezia, Oct 25 2020
a(9) from Neven Juric (neven.juric(AT)apis-it.hr), Jul 11 2005
More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), May 14 2007
Showing 1-5 of 5 results.