A279514 - OEIS

%I #30 Mar 31 2018 16:54:53

%S 1,0,3,0,1,0,12,8,0,9,6,0,0,1,0,96,0,204,0,160,0,67,0,36,0,12,0,0,0,1,

%T 0,2400,1680,480,1824,1200,1300,2300,1600,100,400,400,225,300,70,0,

%U 100,0,0,20,0,0,0,0,1,0,34560,0,87840,0,153840,0,77616,0,61020,0,56048,0,28500,0,9900,0,4075,0,3225,0,1350,0,170,0,225,0,0,0,30,0,0,0,0,0,1,0,2540160,2338560,1058400,1522080,1582560,1225440,1905120,3605616,2342592,1605240

%N Number of terms in the cycle index Z(S_n X S_n) of the Cartesian product of the symmetric group S_n with itself that contain q cycles, where 1 <= q <= n*n. (Triangular array.)

%C A permutation (alpha,beta) from S_n X S_n acts on pairs (p,q) producing (alpha(p), beta(q)) yielding a permutation of the pairs which is factored into cycles to produce the number of cycles. Compare to Stirling numbers of the first kind, which compute the same statistic for Z(S_n).

%D F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, 1973, page 95, section 4.3.

%H Marko R. Riedel, <a href="http://math.stackexchange.com/questions/2056708/">Inequivalent matrices with some number of possible entries under row and column permutation.</a>

%H Marko R. Riedel, <a href="/A279514/a279514.maple.txt">Maple code for cycle index and cycle count.</a>

%e 1

%e 0, 3, 0, 1

%e 0, 12, 8, 0, 9, 6, 0, 0, 1

%e 0, 96, 0, 204, 0, 160, 0, 67, 0, 36, 0, 12, 0, 0, 0, 1

%p seq(CF(q), q=1..7); # CF is defined in the attached Maple file.

%K nonn,tabf

%O 1,3

%A _Marko Riedel_, Dec 13 2016