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Multiplicity-layer decomposition of complete multiset designs

The design whose blocks consist of all $k$-element multisets drawn from a $v$-set, denoted $M(v,k)$, is a classical example of a balanced $(k+1)$-ary design. Although its parameters are well known, existing derivations often rely on general multiset design theory. This paper gives unified elementary derivations of the parameters $b$, $r$, and $\lambda$ using stars-and-bars and double counting. We exhibit a natural multiplicity-layer decomposition: removing $s$ copies of a fixed point from all blocks in which it has multiplicity exactly $s$ yields a family of subdesigns naturally in bijection with $M(v-1,k-s)$. This viewpoint clarifies the recursive structure underlying complete multiset designs. Finally, the multiplicity vectors of blocks of $M(v,k)$ form a $(k+1)$-ary code of length $v$ with constant coordinate sum $k$ and minimum Hamming distance $2$, achieving size $\binom{v+k-1}{k}$.

Combinatorial Press

Apurv Srivastav Sudesh K. Srivastav

Ars Combinatoria

2026

Title: Multiplicity-layer decomposition of complete multiset designs

Description:

Although its parameters are well known, existing derivations often rely on general multiset design theory.

This paper gives unified elementary derivations of the parameters $b$, $r$, and $\lambda$ using stars-and-bars and double counting.

We exhibit a natural multiplicity-layer decomposition: removing $s$ copies of a fixed point from all blocks in which it has multiplicity exactly $s$ yields a family of subdesigns naturally in bijection with $M(v-1,k-s)$.

This viewpoint clarifies the recursive structure underlying complete multiset designs.

Finally, the multiplicity vectors of blocks of $M(v,k)$ form a $(k+1)$-ary code of length $v$ with constant coordinate sum $k$ and minimum Hamming distance $2$, achieving size $\binom{v+k-1}{k}$.

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Multiplicity-layer decomposition of complete multiset designs

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