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The [binary]
ménage problem[1] is concerned with the enumeration of seating arrangements of
opposite-sex couples around a circular table [with
seats], such that
- no pair of adjacent people are of the same sex (FM and MF is allowed; FF and MM is forbidden),
- no spouses sit next to each other,
and where the couples and seats are all labeled.
The [binary]
ménage problem can be restated thus: enumeration of arrangements of integers
to
, such that
- no pair of adjacent numbers may have the same parity,
- neither precedes nor succeeds .
Number of seating arrangements for the ménage problem
The number of seating arrangements
for the ménage problem is
where
is the number of ways of sitting the
ladies first, and
is the number of ways of sitting the
gentlemen thereafter.
A059375 Number of seating arrangements for the ménage problem. [For
opposite-sex couples around a circular table with
seats.] [For
.]
-
{1, 0, 0, 12, 96, 3120, 115200, 5836320, 382072320, 31488549120, 3191834419200, 390445460697600, 56729732529254400, 9659308746908620800, 1905270127543015833600, ...} |
A094047 Number of seating arrangements of
couples around a circular table (up to rotations) so that each person sits between two people of the opposite sex and no couple is seated together. [For
.]
[[[Category:Pages using the math template without the tex argument|Ménage problem]]
]
-
{0, 0, 2, 12, 312, 9600, 416880, 23879520, 1749363840, 159591720960, 17747520940800, 2363738855385600, 371511874881100800, 68045361697964851200, 14367543450324474009600, ...} |
Ménage numbers
The
ménage numbers are the number of ways of sitting the
men after sitting the
ladies. (The ladies can sit in
ways.)
A000179 Ménage numbers: number of permutations
of
such that
and
for all
. [For
.] [[[Category:Pages using the math template without the tex argument|Ménage problem]]
]
-
{1, 0, 0, 1, 2, 13, 80, 579, 4738, 43387, 439792, 4890741, 59216642, 775596313, 10927434464, 164806435783, 2649391469058, 45226435601207, 817056406224416, ...} |
A249510 Ménage primes.
-
{2, 13, 775596313, 567525075138663383127158192994765939404930668817780601409606090331861, ...} |
Ternary ménage problem
The
ternary ménage problem[1] is concerned with the enumeration of seating arrangements of
opposite-sex couples around a circular table [with
seats], such that
- no triple of adjacent people are all of the same sex (FFM, FMF, FMM, MFF, MFM, MMF is allowed; FFF and MMM is forbidden),
- no spouses sit next to each other,
and where the couples and seats are all labeled.
The
ternary ménage problem can be restated thus: enumeration of arrangements of integers
to
, such that
- no triple of adjacent numbers may all have the same parity,
- neither precedes nor succeeds .
Number of seating arrangements for the ternary ménage problem
A258338 Ternary ménage problem: number of seating arrangements for
opposite-sex couples around a circular table such that no spouses and no triples of the same sex sit next to each other.
Seats are labeled. [For
.]
-
{0, 8, 84, 3456, 219120, 19281600, 2324085120, 370554347520, 74897768655360, 18761274367718400, 5708008284647961600, 2072453585852572876800, ...} |
A114939 Number of essentially different [up to rotations and reflections] seating arrangements for
couples around a circular table with
seats avoiding spouses being neighbors and avoiding clusters of 3 persons with equal gender. [For
.] [[[Category:Pages using the math template without the tex argument|Ménage problem]]
]
-
{0, 1, 7, 216, 10956, 803400, 83003040, 11579823360, 2080493573760, 469031859192960, 129727461014726400, 43176116371928601600, 17025803126147196057600, ...} |
Ternary ménage numbers
(...)
Generalized ménage problems
(...)
Notes
- ↑ 1.0 1.1 Cf. slides from Max Alekseyev.
External links