Smooth squared, triangular, and hexagonal bargraphs

Abstract

In this paper, we find an explicit formula for the generating function for the number of smooth squared (triangular, hexagonal) bargraphs according to the perimeter and number of columns. In particular, we show that the number of smooth squared, triangular, and hexagonal bargraphs with perimeter 2n (resp. n, 2n) is asymptotic to csr1?ns/??n3 (resp. ctr1?nt/??n3, ch/??n3?2n+2 ), where rs = 1+ 3?181+24?78/12 ? 23/12 3?181+24?78, rt is the smallest positive root of the polynomial p16?2p14+p12?2p11?2p10+2p9+4p8?5p6?2p5+p4?2p3?2p2+1 and cs, ct are two constants, as n ? ?.