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Furthermore, this study uncovers an inherent symmetry in the distribution of terms, highlighting structured patterns within permanent computation. 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Voor Wiskd."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1080\/03081089508818355","article-title":"On evaluating permanents and a matrix of contangents","volume":"38","author":"Callan","year":"1995","journal-title":"Linear Multilinear Algebra"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"573","DOI":"10.1007\/s10462-012-9354-y","article-title":"A review and analysis of \u201cgraph theoretical-matrix permanent\u201d approach to decision making with example applications","volume":"42","author":"Baykasoglu","year":"2014","journal-title":"Artif. Intell. Rev."},{"key":"ref_11","unstructured":"Cullis, C.E. (1913). Matrices and Determinoids, Cambridge University Press."},{"key":"ref_12","first-page":"17","article-title":"A Definition of the Determinant of a Rectangular Matrix","volume":"1","author":"Radic","year":"1969","journal-title":"Glas. Mat."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1017\/S0004972700011369","article-title":"A determinant for rectangular matrices","volume":"21","author":"Joshi","year":"1980","journal-title":"Bull. Aust. Math. Soc."},{"key":"ref_14","first-page":"29","article-title":"M\u00e9moire sur un syst\u00e8me de formules analytiques et leur application \u00e0 des consid\u00e9rations g\u00e9om\u00e9triques","volume":"10","author":"Binet","year":"1813","journal-title":"J. Ec. Polytech."},{"key":"ref_15","unstructured":"Rota, G.C. (1978). Permanents. Encyclopedia of Mathematics and its Applications, Addison-Wesley Publishing."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Ryser, H.J. (1963). Combinatorial Mathematics, The Mathematical Association of America. The Carus Mathematical Monographs, No. 14.","DOI":"10.5948\/UPO9781614440147"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1017\/S0013091500027760","article-title":"Evaluation of Permanents","volume":"22","author":"Minc","year":"1979","journal-title":"Proc. Edinb. Math. Soc."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1","DOI":"10.2140\/pjm.1982.101.1","article-title":"On the evaluation of permanents","volume":"101","author":"Bebiano","year":"1982","journal-title":"Pac. J. Math."},{"key":"ref_19","unstructured":"Weisstein, E.W. (2025, March 25). Subset Sum Problem. Available online: https:\/\/mathworld.wolfram.com\/SubsetSumProblem.html."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"721","DOI":"10.1080\/00029890.1982.11995524","article-title":"Solution of two difficult combinatorial problems with linear algebra","volume":"89","author":"Proctor","year":"1982","journal-title":"Am. Math. 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