SMOOTH SQUARED, TRIANGULAR, AND HEXAGONAL BARGRAPHS

In this paper, we find an explicit formula for the generating function for thenumber of smooth squared (triangular, hexagonal) bargraphs according to theperimeter and number of columns. In particular, we show that the numberof smooth squared, triangular, and hexagonal bargraphs with perimeter 2n(resp.n, 2n) is asymptotic to c(s)r(s)(1-n)/root pi n(3)(resp.c(t)r(t)(1-n)/root pi n(3),c(h)root pi n(3)/root 2(n+2)), wherers=1+3 root 181+24 root 78/12-23/12(3)root 181+24 root 78,rtis the smallest positive root of thepolynomial p(16)-2p(14)+p(12)-2p(11)-2p(10)+2p(9)+4p(8)-5p(6)-2p(5)+p(4)-2p(3)-2p(2)+1and c(s), c(t), c(h) are three constants, asn7 ->infinity.