Geodesic
Generalized valences
Matematičeskie zametki, Tome 17 (1975) no. 3, pp. 433-442 Cet article a éte moissonné depuis la source Math-Net.Ru

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We have established that $V(S_p,q;G)$, namely, the collection of all those edges of an arbitrary $n$-vertex hypergraph $G$, whose intersections with set $S_p$, $p$ vertices, has a cardinality $q$, satisfies certain identity relations; in particular, if $v(S_p,q;G)=|V(S_p,q;G)|$, then $$ v(S_p,q;G)=\sum_{i\ge0}(-1)^iC_{q+1}^q\sum_{S_{q+i}\subset S_p}v(S_{q+i},q+i;G). $$ As applications we derive two new combinatorial identities. 
@article{MZM_1975_17_3_a9,
     author = {B. S. Stechkin},
     title = {Generalized valences},
     journal = {Matemati\v{c}eskie zametki},
     pages = {433--442},
     year = {1975},
     volume = {17},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1975_17_3_a9/}
}
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B. S. Stechkin. Generalized valences. Matematičeskie zametki, Tome 17 (1975) no. 3, pp. 433-442. http://geodesic.mathdoc.fr/item/MZM_1975_17_3_a9/