Upper bounds for permanents of $\left( {0,\,1} \right)$-matrices

Minc, Henryk

doi:10.1090/s0002-9904-1963-11031-9

Cited by 100 publications

(55 citation statements)

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“…It was conjectured by Mine [3] and proved by Brègman [1] that if A is a (0, l)-matrix of order n with row sums ri,r"2,... ,rn, then where Sn is the symmetric group on {1,2,...,n}. We now state our principal result.…”

Section: Introductionmentioning

confidence: 83%

An upper bound for the permanent of a 3-dimensional (0,1)-matrix

Dow¹,

Gibson²

1987

Proc. Amer. Math. Soc.

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Let A = ( a i j k ) A = ({a_{ijk}}) be a 3 3 -dimensional matrix of order n n . The permanent of A A is defined by \[ per ⁡ A = ∑ σ , τ ∈ S n ∏ i = 1 n a i σ ( i ) τ ( i ) , \operatorname {per} A = \sum \limits _{\sigma ,\tau \in {S_n}} {\prod \limits _{i = 1}^n {{a_{i\sigma (i)\tau (i)}},} } \] where S n {S_n} is the symmetric group on { 1 , 2 , … , n } \{ 1,2, \ldots ,n\} . Suppose that A A is a (0,1)-matrix and that r i = ∑ j , k = 1 n a i j k for i = 1 , 2 , … , n {r_i} = \sum \nolimits _{j,k = 1}^n {{a_{ijk}}} {\text { for }}i = 1,2, \ldots ,n . In this paper it is shown that per A ≤ ∏ i = 1 n r i ! 1 / r i . A \leq \prod \nolimits _{i = 1}^n {{r_i}{!^{1/{r_i}}}.} A similar bound is then obtained for a second function, the 2 2 -permanent of a 3 3 -dimensional matrix, that is another analogue of the permanent of an ordinary ( 2 2 -dimensional) matrix.

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Section: Introductionmentioning

confidence: 83%

An upper bound for the permanent of a 3-dimensional (0,1)-matrix

Dow¹,

Gibson²

1987

Proc. Amer. Math. Soc.

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“…); this will follow from the improved inequality (3) [El => 6 log6 for an arbitrary (0, 1)-matrix M was conjectured by Minc [3] and proved by Bregman [2] (see Schrijver [6] for a particularly simple and elegant proof). Since the expression…”

Section: ])mentioning

confidence: 97%

A New Lower Bound for the Number of Switches in Rearrangeable Networks

Pippenger¹

1980

SIAM. J. on Algebraic and Discrete Methods

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“…For the number of perfect matchings, the bipartite case was conjectured in 1963 by Minc [47] and proved by Brègman [5] a decade later. Many different proofs have been given since then.…”

Section: Theorem 99 ([36]mentioning

confidence: 99%