Bounded compositions and products of chains, or bi(s)nomials everywhere from de Moivre's formula to spin multiplicities and constant-sum codes

Hacène Belbachir, José Ricardo G. Mendonça

How many ways are there to select $k$ objects from $n$ types when type $i$ is available in only $s_i$ copies? For equal bounds, this is de Moivre's problem of obtaining a prescri…