We explore how the asymptotic structure of a random $n$-term weak integer composition of $m$ evolves, as $m$ increases from zero. The primary focus is on establishing thresholds for the appearance and disappearance of substructures. These include the longest and shortest runs of zero terms or of nonzero terms, longest increasing runs, longest runs of equal terms, largest squares (runs of $k$ terms each equal to $k$), as well as a wide variety of other patterns. Of particular note is the dichotomy between the appearance and disappearance of exact consecutive patterns, with smaller patterns appearing before larger ones, whereas longer patterns disappear before shorter ones.
@article{10_37236_13010,
author = {David Bevan and Dan Threlfall},
title = {On the evolution of random integer compositions},
journal = {The electronic journal of combinatorics},
year = {2025},
volume = {32},
number = {1},
doi = {10.37236/13010},
zbl = {1559.60024},
url = {http://geodesic.mathdoc.fr/articles/10.37236/13010/}
}
TY - JOUR
AU - David Bevan
AU - Dan Threlfall
TI - On the evolution of random integer compositions
JO - The electronic journal of combinatorics
PY - 2025
VL - 32
IS - 1
UR - http://geodesic.mathdoc.fr/articles/10.37236/13010/
DO - 10.37236/13010
ID - 10_37236_13010
ER -
%0 Journal Article
%A David Bevan
%A Dan Threlfall
%T On the evolution of random integer compositions
%J The electronic journal of combinatorics
%D 2025
%V 32
%N 1
%U http://geodesic.mathdoc.fr/articles/10.37236/13010/
%R 10.37236/13010
%F 10_37236_13010
David Bevan; Dan Threlfall. On the evolution of random integer compositions. The electronic journal of combinatorics, Tome 32 (2025) no. 1. doi: 10.37236/13010
