We provide a combinatorial formula for the expansion of immaculate noncommutative symmetric functions into complete homogeneous noncommutative symmetric functions. To do this, we introduce generalizations of Ferrers diagrams which we call GBPR diagrams. A GBPR diagram assigns a color (grey, blue, purple, or red) to each cell of the diagram. We define tunnel hooks, which play a role similar to that of the special rim hooks appearing in the Eğecioğlu-Remmel formula for the symmetric inverse Kostka matrix. We extend this interpretation to skew shapes and fully generalize to define immaculate functions indexed by integer sequences skewed by integer sequences. Finally, as an application of our combinatorial formula, we extend Campbell's results on ribbon decompositions of immaculate functions to a larger class of shapes.
@article{10_37236_12677,
author = {Edward Allen and Sarah Katherine Mason},
title = {A combinatorial interpretation of the noncommutative inverse {Kostka} matrix},
journal = {The electronic journal of combinatorics},
year = {2025},
volume = {32},
number = {1},
doi = {10.37236/12677},
zbl = {1560.05130},
url = {http://geodesic.mathdoc.fr/articles/10.37236/12677/}
}
TY - JOUR
AU - Edward Allen
AU - Sarah Katherine Mason
TI - A combinatorial interpretation of the noncommutative inverse Kostka matrix
JO - The electronic journal of combinatorics
PY - 2025
VL - 32
IS - 1
UR - http://geodesic.mathdoc.fr/articles/10.37236/12677/
DO - 10.37236/12677
ID - 10_37236_12677
ER -
%0 Journal Article
%A Edward Allen
%A Sarah Katherine Mason
%T A combinatorial interpretation of the noncommutative inverse Kostka matrix
%J The electronic journal of combinatorics
%D 2025
%V 32
%N 1
%U http://geodesic.mathdoc.fr/articles/10.37236/12677/
%R 10.37236/12677
%F 10_37236_12677
Edward Allen; Sarah Katherine Mason. A combinatorial interpretation of the noncommutative inverse Kostka matrix. The electronic journal of combinatorics, Tome 32 (2025) no. 1. doi: 10.37236/12677
