Columns of fixed height in bargraphs
نویسندگان
1 The John Knopfmacher Centre for Applicable Analysis and Number Theory, School of Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050,Johannesburg, South Africa
2 The John Knopfmacher Centre for Applicable Analysis and Number Theory, School of Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050,Johannesburg, South Africa
3 The John Knopfmacher Centre for Applicable Analysis and Number Theory, School of Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050,Johannesburg, South Africa
doi
10.22108/toc.2023.132462.1957چکیده
We 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 $r$.
