Signed enumeration of upper-right corners in path shuffles

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• 作者：William Kuszmaulp>1p>p> ; p>p>william.kuszmaul@gmail.com" class="auth_mail" title="E-mail the corresponding authorp>
• 刊名：European Journal of Combinatorics
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：60
• 期：Complete
• 页码：100-113
• 全文大小：431 K

文摘

We resolve a conjecture of Albert and Bousquet-Mélou enumerating quarter-planar walks with fixed horizontal and vertical projections according to their upper-right-corner count modulo 2. In doing this, we introduce a signed upper-right-corner count statistic. We find its distribution over planar walks with any choice of fixed horizontal and vertical projections. Additionally, we prove that the polynomial counting loops with a fixed horizontal and vertical projection according to the absolute value of their signed upper-right-corner count is pan id="mmlsi1" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0195669816300841&_mathId=si1.gif&_user=111111111&_pii=S0195669816300841&_rdoc=1&_issn=01956698&md5=2f0ab6d494716e8125977abb48298c30" title="Click to view the MathML source">(x+1)pan>pan class="mathContainer hidden">pan class="mathCode">$(x+1)$pan>pan>pan>-positive. Finally, we conjecture an equivalence between pan id="mmlsi1" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0195669816300841&_mathId=si1.gif&_user=111111111&_pii=S0195669816300841&_rdoc=1&_issn=01956698&md5=2f0ab6d494716e8125977abb48298c30" title="Click to view the MathML source">(x+1)pan>pan class="mathContainer hidden">pan class="mathCode">$(x+1)$pan>pan>pan>-positivity of the generating function for upper-right-corner count and signed upper-right-corner count, leading to a reformulation of a conjecture of Albert and Bousquet-Mélou on which their asymptotic analysis of permutations is sortable by two stacks in parallel relies.