Interspersion

An Array $A=a_{ij}$, $i,j\geq 1$ of Positive Integers is called an interspersion if

1. The rows of $A$ comprise a Partition of the Positive Integers,

2. Every row of $A$ is an increasing sequence,

3. Every column of $A$ is a (possibly Finite) increasing sequence,

4. If $(u_j)$ and $(v_j)$ are distinct rows of $A$ and if $p$ and $q$ are any indices for which $u_p<v_q<u_{p+1}$, then $u_{p+1}<v_{q+1}<u_{p+2}$.

If an array $A=a_{ij}$ is an interspersion, then it is a Dispersion. If an array $A=a(i,j)$ is an interspersion, then the sequence $\{x_n\}$ given by $\{x_n=i: n=(i,j)\}$ for some $j$ is a Fractal Sequence. Examples of interspersion are the Stolarsky Array and Wythoff Array.

See also Dispersion (Sequence), Fractal Sequence, Stolarsky Array

References

Kimberling, C. ``Interspersions and Dispersions.'' Proc. Amer. Math. Soc. 117, 313-321, 1993.

Kimberling, C. ``Fractal Sequences and Interspersions.'' Ars Combin. 45, 157-168, 1997.