Geodesic
Formalization of combinatorial numbers in terms of entire solutions of systems of linear Diophantine equations
Prikladnaya Diskretnaya Matematika. Supplement, no. 7 (2014), pp. 157-160.

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Some generalizations of the number of placements with repetitions and various restrictions are considered. Counting these combinatorial numbers leads to the definition of nonnegative solutions of systems of linear Diophantine equations under appropriate additional restrictions. Generating functions and integral formulas for calculating input combinatorial numbers are obtained, and various problems that are solved with their application are discussed.
Keywords: combinatorial numbers, systems of linear Diophantine equations, generating functions. 
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     author = {V. V. Gotsulenko},
     title = {Formalization of combinatorial numbers in terms of entire solutions of systems of linear {Diophantine} equations},
     journal = {Prikladnaya Diskretnaya Matematika. Supplement},
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V. V. Gotsulenko. Formalization of combinatorial numbers in terms of entire solutions of systems of linear Diophantine equations. Prikladnaya Diskretnaya Matematika. Supplement, no. 7 (2014), pp. 157-160. http://geodesic.mathdoc.fr/item/PDMA_2014_7_a66/

[1] Yablonskii S. V., Vvedenie v diskretnuyu matematiku, Nauka, M., 1986, 384 pp. | MR

[2] Gotsulenko V. V., “Formula dlya chisla sochetanii s povtoreniyami pri ogranicheniyakh i eë primenenie”, Prikladnaya diskretnaya matematika, 2013, no. 2(20), 71–77