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A279514

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.)

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

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OFFSET

1,3

COMMENTS

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).

REFERENCES

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

LINKS

Table of n, a(n) for n=1..102.

Marko R. Riedel, Inequivalent matrices with some number of possible entries under row and column permutation.

Marko R. Riedel, Maple code for cycle index and cycle count.

EXAMPLE

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

MAPLE

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