University of IsfahanTransactions on Combinatorics2251-865713120240301Columns of fixed height in bargraphs67842719410.22108/toc.2023.132462.1957ENMargaretArchibaldThe John Knopfmacher Centre for Applicable Analysis and Number Theory, School of Mathematics, University of the
Witwatersrand, Private Bag 3, Wits 2050,Johannesburg, South Africa0000-0001-5635-6733AubreyBlecherThe John Knopfmacher Centre for Applicable Analysis and Number Theory, School of Mathematics, University of the
Witwatersrand, Private Bag 3, Wits 2050,Johannesburg, South Africa0000-0003-2487-3220ArnoldKnopfmacherThe John Knopfmacher Centre for Applicable Analysis and Number Theory, School of Mathematics, University of the
Witwatersrand, Private Bag 3, Wits 2050,Johannesburg, South Africa0000-0003-1962-043XJournal Article20220124We obtain the generating function for the number of columns of fixed height $r$ in a bargraph (classified according to semi-perimeter). As initial case for two distinct methods we first find the generating function for columns of height $1$. Then using a first-return-to-level-$1$ decomposition, we obtain the rational function version of the continued fraction generating function which allows us to derive separate recursions for its numerator and denominator. This then allows us to get the asymptotic average number of columns for each $r$. We also obtain an equivalent generating function by exploiting a sequential decomposition for bargraphs in terms of columns of height $r$.https://toc.ui.ac.ir/article_27194_6e4c52901f9d4f87e347e0a976d4aeed.pdf