Geodesic
The listing and counting combinatorial algorithm for compositions of a natural number with constraints
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the 6th International Conference "Dynamic Systems and Computer Science: Theory and Applications" (DYSC 2024). Irkutsk, September 16-20, 2024. Part 2, Tome 239 (2025), pp. 13-24

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In this paper, we propose a listing and counting algorithm for compositions of a natural number based on combinatorial objects of a hierarchical structure, such as Pascal's triangle, Pascal's pyramid, and Pascal's hyperpyramids. We obtain the recurrent relation that is the basis for listing and counting of compositions of a natural number with an arbitrary constraints on the values of its natural parts and the formula for explicit counting of compositions and a generating function for the number of compositions.
Keywords: composition of number, recurrence relation, generating function, Fibonacci numbers, Tribonacci numbers, Tetranacci numbers, Pentanacci numbers
Mots-clés : Pascal's hyperpyramid, Pascal's pyramid, Pascal's triangle, polynomial coefficients, trinomial coefficients, binomial coefficients 
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     author = {O. V. Kuz'min and M. V. Strihar},
     title = {The listing and counting combinatorial algorithm for compositions of a natural number with constraints},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {13--24},
     publisher = {mathdoc},
     volume = {239},
     year = {2025},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2025_239_a1/}
}
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O. V. Kuz'min; M. V. Strihar. The listing and counting combinatorial algorithm for compositions of a natural number with constraints. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the 6th International Conference "Dynamic Systems and Computer Science: Theory and Applications" (DYSC 2024). Irkutsk, September 16-20, 2024. Part 2, Tome 239 (2025), pp. 13-24. http://geodesic.mathdoc.fr/item/INTO_2025_239_a1/