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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
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 (list; graph; refs; listen; history; text; internal format)
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

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

MAPLE

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

CROSSREFS

Sequence in context: A210473 A185951 A188832 * A094675 A200472 A263753

Adjacent sequences:  A279511 A279512 A279513 * A279515 A279516 A279517

KEYWORD

nonn,tabf

AUTHOR

Marko Riedel, Dec 13 2016

STATUS

approved

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Last modified July 26 06:03 EDT 2017. Contains 289798 sequences.