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Van der Waerden's permanent conjecture
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(Theorem)
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Let  be any doubly stochastic  matrix (i.e. nonnegative real entries, each row sums to 1, each column too, hence square).
Let  be the one where all entries are equal (i.e. they are
 ). Its permanent works out to
and Van der Waerden conjectured in 1926 that this is the smallest value for the permanent of any doubly stochastic  , and is attained only for  :
It was finally proven independently by Egorychev and by Falikman, in 1979/80.
- Hal86
- MARSHALL J. HALL, JR., Combinatorial Theory (2nd ed.),
Wiley 1986, repr. 1998, ISBN0471091383 and 0471315184
has a proof of the permanent conjecture.
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"Van der Waerden's permanent conjecture" is owned by marijke.
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(view preamble)
| Other names: |
permanent conjecture |
| Keywords: |
doubly stochastic matrix |
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Cross-references: conjecture, permanent, real, matrix, doubly stochastic
There is 1 reference to this entry.
This is version 2 of Van der Waerden's permanent conjecture, born on 2005-04-08, modified 2005-04-08.
Object id is 6935, canonical name is VanDerWaerdensPermanentConjecture.
Accessed 2298 times total.
Classification:
| AMS MSC: | 15A51 (Linear and multilinear algebra; matrix theory :: Stochastic matrices) | | | 15A15 (Linear and multilinear algebra; matrix theory :: Determinants, permanents, other special matrix functions) |
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Pending Errata and Addenda
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