COUNTING ROOTED EULERIAN MAPS ON THE PROJECTIVE PLANE

It is well known that any kind of exact enumerations of rooted maps on nonplanar surface is quite difficult. This paper presents a functional equation for rooted Eulerian maps on the projective plane. A parametric expression, through which an exact enumeration by root-valency and the number of edges of maps may be determined, is obtained.

ON THE VERTEX PARTITION EQUATION OF ROOTED LOOPLESS PLANAR MAPS

This paper provides a functional equation astisfied by the generating function for enumerating rooted loopless planar maps with vertex partition. A kind of applications in enumerating, by providing explicit formulae, a type of rooted loopless planar maps with the maximum valency of vertices given are described. Meanwhile, the functional equation for enumerating rooted loopless planar maps (connected) with the edge number and the valency of root-vertex as the parameters is also derived directly.

Genetic dissection of maize seedling root system architecture traits using an ultra-high density bin-map and a recombinant inbred line population

Maize（Zea mays） root system architecture（RSA）mediates the key functions of plant anchorage and acquisition of nutrients and water. In this study,a set of 204 recombinant inbred lines（RILs） was derived from the widely adapted Chinese hybrid ZD958（Zheng58 Chang7-2）,genotyped by sequencing（GBS） and evaluated as seedlings for 24 RSA related traits divided into primary,seminal and total root classes. Signi ficant differences between the means of the parental phenotypes were detected for 18 traits,and extensive transgressive segregation in the RIL population was observed for all traits. Moderate to strong relationships among the traits were discovered. A total of 62 quantitative trait loci（QTL） were identi fied that individually explained from1.6% to 11.6%（total root dry weight/total seedling shoot dry weight） of the phenotypic variation. Eighteen,24 and 20 QTL were identi fied for primary,seminal and total root classes of traits,respectively. We found hotspots of 5,3,4 and 12 QTL in maize chromosome bins 2.06,3.02-03,9.02-04,and 9.05-06,respectively,implicating the presence of root gene clusters or pleiotropic effects. These results characterized the phenotypic variation and genetic architecture of seedling RSA in a population derived from a successful maize hybrid.

The Number of Rooted Circuit Boundary Maps

In this article, the authors discuss two kinds of new planar maps： pan-fan maps and circuit boundary maps, and provide explicit expressions about their enumerating functions with different parameters. Meanwhile, two explicit counting formulas for circuit cubic boundary maps with two parameters; the size and the valency of the root-face, are also extracted.

ENUMERATION OF ROOTED NONSEPARABLE OUTERPLANAR MAPS

In this paper, the number of combinatorially distinct rooted nonseparable outerplanar maps withm edges and the valency of the root-face being n is found to be(m-1)! (m-2) !:(n-1)!(n-2)! (m-n)!(m-n+1)!and, the number of rooted nonseparable outerplanar maps with m edges is also determined to be(2m-2)!:(m-1)!m!,which is just the number of distinct rooted plane trees with m-1 edges.

A Census of Petal Bundles by Genus

This paper provides the uniform enumerative functional equation of orientable （nonori-entable） rooted petal bundles with more parameters, and deduces two recursion formulas for calculation. Accordingly, an explicit expression of rooted petal bundles with up to two parameters on nonorientable surface of genus 4 is also obtained.