If your socks come out of the laundry all mixed up, how should you sort them? We introduce and study a novel foot-sorting algorithm that uses feet to attempt to sort a sock ordering; one can view this algorithm as an analogue of Knuth's stack-sorting algorithm for set partitions. The sock orderings that can be sorted using a fixed number of feet are characterized by Klazar's notion of set partition pattern containment. We give an enumeration involving Fibonacci numbers for the $1$-foot-sortable sock orderings within a naturally-arising class. We also prove that if you have socks of $n$ different colors, then you can always sort them using at most $\left\lceil\log_2(n)\right\rceil$ feet, and we use a Ramsey-theoretic argument to show that this bound is tight.
@article{10_37236_11934,
author = {Colin Defant and Noah Kravitz},
title = {Foot-sorting for socks},
journal = {The electronic journal of combinatorics},
year = {2024},
volume = {31},
number = {3},
doi = {10.37236/11934},
zbl = {1548.05021},
url = {http://geodesic.mathdoc.fr/articles/10.37236/11934/}
}
TY - JOUR
AU - Colin Defant
AU - Noah Kravitz
TI - Foot-sorting for socks
JO - The electronic journal of combinatorics
PY - 2024
VL - 31
IS - 3
UR - http://geodesic.mathdoc.fr/articles/10.37236/11934/
DO - 10.37236/11934
ID - 10_37236_11934
ER -
%0 Journal Article
%A Colin Defant
%A Noah Kravitz
%T Foot-sorting for socks
%J The electronic journal of combinatorics
%D 2024
%V 31
%N 3
%U http://geodesic.mathdoc.fr/articles/10.37236/11934/
%R 10.37236/11934
%F 10_37236_11934
Colin Defant; Noah Kravitz. Foot-sorting for socks. The electronic journal of combinatorics, Tome 31 (2024) no. 3. doi: 10.37236/11934
