# zbMATH — the first resource for mathematics

Permanental compounds and permanents of (0,1)-circulants. (English) Zbl 0608.15009
Author’s summary: It was shown by the author in a recent paper that a recurrence relation for permanents of (0,1)-circulants can be generated from the product of the characteristic polynomials of permanental compounds of the companion matrix of a polynomial associated with (0,1)- circulants of the given type. In the present paper general properties of permanental compounds of companion matrices are studied, and in particular of convertible companion matrices, i.e., matrices whose permanental compounds are equal to the determinantal compounds after changing the signs of some of their entries. These results are used to obtain formulas for the limit of the nth root of the permanent of the $$n\times n$$ (0,1)-circulant of a given type, as n tends to infinity. The root-squaring method is then used to evaluate this limit for a wide range of circulant types whose associated polynomials have convertible companion matrices.
Reviewer: N J.Pullman

##### MSC:
 15B36 Matrices of integers 15A15 Determinants, permanents, traces, other special matrix functions 05B20 Combinatorial aspects of matrices (incidence, Hadamard, etc.)
Full Text:
##### References:
 [1] Hammersley, J.M., Existence theorems and Monte Carlo methods for the monomer-dimer problem, Research papers in statistics: festschrift for J. Neyman, 125-146, (1966) · Zbl 0161.15401 [2] Hammersley, J.M., An improved lower bound for the multidimensional dimer problem, Proc. Cambridge philos. soc., 64, 455-463, (1968) · Zbl 0155.03002 [3] Marcus, M.; May, F.C., The permanent function, Canad. J. math., 14, 177-189, (1962) · Zbl 0106.01601 [4] Marcus, M.; Minc, H., On the relation between the determinant and the permanent, Illinois J. math., 5, 376-381, (1961) · Zbl 0104.00904 [5] Marcus, M.; Minc, H., A survey of matrix theory and matrix inequalities, (1964), Allyn and Bacon Boston · Zbl 0126.02404 [6] Metropolis, N.; Stein, M.L.; Stein, P.R., Permanents of cyclic (0, 1) matrices, J. combin. theory, 7, 291-321, (1969) · Zbl 0183.29803 [7] Minc, H., An upper bound for the multidimensional dimer problem, Proc. Cambridge philos. soc., 83, 461-462, (1978) · Zbl 0383.05005 [8] Minc, H., Permanents, () · Zbl 0166.29904 [9] Minc, H., Theory of permanents 1978-1981, Linear and multilinear algebra, 12, 227-263, (1983) · Zbl 0511.15002 [10] Minc, H., Recurrence formulas for permanents of (0, 1)-circulants, Linear algebra appl., 71, 241-265, (1985) · Zbl 0578.15011 [11] Schrijver, A.; Valiant, W.G., On lower bounds for permanents, Proc. kon. nederl. akad. wetensch. A83 = indag. math., 42, 425-427, (1980) · Zbl 0451.15009 [12] Whittaker, E.T.; Robinson, G., The calculus of observations, (1940), Blackie London
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.