Census of Planar Maps: From the One-Matrix Model Solution to a Combinatorial Proof

We consider the problem of enumeration of planar maps and revisit its one-matrix model solution in the light of recent combinatorial techniques involving conjugated trees. We adapt and generalize these techniques so as to give an alternative and purely combinatorial solution to the problem of counting arbitrary planar maps with prescribed vertex degrees.Comment: 29 pages, 14 figures, tex, harvmac, eps

Supplementary Material for: Dermoscopic Features of Subungual Squamous Cell Carcinoma: A Study of 44 Cases

<i>Background:</i> Subungual squamous cell carcinoma (SSCC) is the most frequent tumor of the nail apparatus. Its diagnosis is often missed or delayed because the clinical presentation is atypical and can mimic other conditions. Accurate diagnosis can only be made by performing an appropriate surgical biopsy, but biopsy is painful and often leaves definitive dystrophic scars. The use of dermoscopy, a noninvasive technique, has been described to be useful for the preoperative evaluation of nail diseases. <i>Objectives:</i> To define the different clinical and dermoscopic presentations of SSCC and to compare them with onychomatricoma-associated clinical and dermoscopic features published in our previous study. <i>Methods:</i> A retrospective review of 44 cases of SSCC seen in our institution over an 8-year period. Six observers scored 19 clinical criteria and 14 dermoscopic criteria as present or absent. Then, we compared those data to a previously published study about the preoperative diagnosis of onychomatricoma. <i>Results:</i> Only 1 dermoscopic criterion was significantly associated with SSCC compared to onychomatricoma: localized hyperkeratosis (odds ratio, OR = 6.25, <i>p</i> = 0.012, 95% confidence interval CI = 1.50-26.01). In contrast, parallel edges (OR = 0.03, <i>p</i> < 0,001, 95% CI = 0.003-0.20) and sharp demarcation of the lesion (OR = 0.24, <i>p</i> = 0.004, 95% CI = 0.09-0.63) can statistically significantly be considered as in favor of onychomatricoma. By contrast, we believe that the presence of unparalleled lateral edges of the nail lesion or of fuzzy edges are more in favor of SSCC. <i>Conclusions:</i> Dermoscopy of the nail plate and of the nail free edge in SSCC provides useful information in order to better select cases to be submitted to biopsy