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A285522 Array read by antidiagonals: T(m,n) = number of circulant digraphs up to Cayley isomorphism on n vertices with edges colored according to step value using a maximum of m-1 colors.

%I A285522 #34 Feb 16 2025 08:33:44
%S A285522 1,1,1,1,2,1,1,3,3,1,1,6,6,4,1,1,6,18,10,5,1,1,20,24,40,15,6,1,1,14,
%T A285522 135,70,75,21,7,1,1,48,130,544,165,126,28,8,1,1,52,648,700,1625,336,
%U A285522 196,36,9,1,1,140,1137,4480,2635,3996,616,288,45,10,1
%N A285522 Array read by antidiagonals: T(m,n) = number of circulant digraphs up to Cayley isomorphism on n vertices with edges colored according to step value using a maximum of m-1 colors.
%C A285522 For the base case of m=2 the sequence counts circulant digraphs up to Cayley isomorphism. Two circulant graphs are Cayley isomorphic if there is a d, which is necessarily prime to n, that transforms through multiplication modulo n the step values of one graph into those of the other. For squarefree n this is the only way that two circulant graphs can be isomorphic. (See Liskovets reference for a proof.)
%C A285522 Alternatively, the number of mappings with domain {1..n-1} and codomain {1..m} up to equivalence. Mappings A and B are equivalent if there is a d, prime to n, such that A(i) = B(i*d mod n) for i in {1..n-1}. This sequence differs from A132191 only in that sequence also includes 0 in the domain which introduces an extra factor of m into the results since zero multiplied by anything is zero.
%C A285522 All column sequences are polynomials of order n-1 and these are the cycle index polynomials.
%C A285522 This sequence is also related to A075195(n, m) which counts necklaces and A285548(m, n) which is the sequence described in the Titsworth reference. In particular, A075195 is the analogous array with equivalence determined through the additive group instead of by multiplication whereas A285548 allows for both addition and multiplication.
%H A285522 Andrew Howroyd, <a href="/A285522/b285522.txt">Table of n, a(n) for n = 1..1275</a>
%H A285522 V. A. Liskovets and R. Poeschel, <a href="https://citeseerx.ist.psu.edu/pdf/b76573e0c2df2ff117cef015809e232a3747f585">On the enumeration of circulant graphs of prime-power and squarefree orders.</a>
%H A285522 R. C. Titsworth, <a href="http://projecteuclid.org/euclid.ijm/1256059671">Equivalence classes of periodic sequences</a>, Illinois J. Math., 8 (1964), 266-270.
%H A285522 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CirculantGraph.html">Circulant Graph.</a>
%H A285522 Wikipedia, <a href="https://en.wikipedia.org/wiki/P%C3%B3lya_enumeration_theorem">Polya enumeration theorem.</a>
%F A285522 T(m, n) = A132191(m, n) / m.
%e A285522 Table starts:
%e A285522 \n  1 2  3   4    5    6    7     8      9      10
%e A285522 m\ ---------------------------------------------------
%e A285522 1 | 1 1  1   1    1    1    1     1      1       1 ...
%e A285522 2 | 1 2  3   6    6   20   14    48     52     140 ...
%e A285522 3 | 1 3  6  18   24  135  130   648   1137    4995 ...
%e A285522 4 | 1 4 10  40   70  544  700  4480  11056   65824 ...
%e A285522 5 | 1 5 15  75  165 1625 2635 20625  65425  489125 ...
%e A285522 6 | 1 6 21 126  336 3996 7826 72576 280596 2521476 ...
%e A285522 ...
%e A285522 Case n=10:
%e A285522 Only 1, 3, 7, 9 are prime to 10.
%e A285522 Multiplication modulo 10 is described by the following multiplication table.
%e A285522   1, 2, 3, 4, 5, 6, 7, 8, 9  => (1)(2)(3)(4)(5)(6)(7)(8)(9) => m^9
%e A285522   3, 6, 9, 2, 5, 8, 1, 4, 7  => (1397)(2684)(5)             => m^3
%e A285522   7, 4, 1, 8, 5, 2, 9, 6, 3  => (1793)(2486)(5)             => m^3
%e A285522   9, 8, 7, 6, 5, 4, 3, 2, 1  => (19)(28)(37)(46)(5)         => m^5
%e A285522 Each row of the multiplication table can be viewed as a permutation and together these form a commutative group on 4 elements. In this case the group is isomorphic to the cyclic group C_4. Each permutation can be represented in cycle notation. (shown above to the right of the corresponding multiplication table row). In order to count the equivalence classes using Polya's enumeration theorem only the number of cycles in each permutation is needed.
%e A285522 This gives the cycle index polynomial (1/4)*(m^9 + m^5 + 2*m^3). Putting m = 1..4 gives 1, 140, 4995, 65824.
%t A285522 A132191[m_, n_] := (1/EulerPhi[n])*Sum[If[GCD[k, n] == 1, m^DivisorSum[n, EulerPhi[#] / MultiplicativeOrder[k, #] &], 0], {k, 1, n}];
%t A285522 T[m_, n_] := A132191[m, n]/m;
%t A285522 Table[T[m - n + 1, n], {m, 1, 11}, {n, m, 1, -1}] // Flatten (* _Jean-François Alcover_, Jun 06 2017 *)
%o A285522 (PARI)
%o A285522 a(n,x)=sum(k=1, n, if(gcd(k, n)==1, x^(sumdiv(n, d, eulerphi(d)/znorder(Mod(k, d)))-1), 0))/eulerphi(n);
%o A285522 for(m=1, 6, for(n=1, 10, print1( a(n,m), ", ") ); print(); );
%Y A285522 Row 2 is A056391.
%Y A285522 Columns include A002411, A006528, A168178, A006565, A282613, A054624.
%Y A285522 Cf. A132191, A075195, A285548, A002729.
%K A285522 nonn,tabl
%O A285522 1,5
%A A285522 _Andrew Howroyd_, Apr 20 2017