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• W2963859359 abstract "We present upper and lower bounds for the number <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=i Subscript n> <mml:semantics> <mml:msub> <mml:mi>i</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:annotation encoding=application/x-tex>i_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of interval graphs on <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=n> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding=application/x-tex>n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> vertices. Answering a question posed by Hanlon, we show that the ordinary generating function <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper I left-parenthesis x right-parenthesis equals sigma-summation Underscript n greater-than-or-equal-to 0 Endscripts i Subscript n Baseline x Superscript n> <mml:semantics> <mml:mrow> <mml:mi>I</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>=</mml:mo> <mml:munder> <mml:mo>∑<!-- ∑ --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>n</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:munder> <mml:msub> <mml:mi>i</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mspace width=thinmathspace /> <mml:msup> <mml:mi>x</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>I(x) = sum _{nge 0} i_n,x^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for the number <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=i Subscript n> <mml:semantics> <mml:msub> <mml:mi>i</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:annotation encoding=application/x-tex>i_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=n> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding=application/x-tex>n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-vertex interval graphs has radius of convergence zero. We also show that the exponential generating function <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper J left-parenthesis x right-parenthesis equals sigma-summation Underscript n greater-than-or-equal-to 0 Endscripts i Subscript n Baseline x Superscript n Baseline slash n factorial> <mml:semantics> <mml:mrow> <mml:mi>J</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>=</mml:mo> <mml:munder> <mml:mo>∑<!-- ∑ --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>n</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:munder> <mml:msub> <mml:mi>i</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mspace width=thinmathspace /> <mml:msup> <mml:mi>x</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>n</mml:mi> <mml:mo>!</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>J(x) = sum _{nge 0} i_n,x^n/n!</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has radius of convergence at least <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=1 slash 2> <mml:semantics> <mml:mrow> <mml:mn>1</mml:mn> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>1/2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>." @default.
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• W2963859359 date "2017-02-24" @default.
• W2963859359 modified "2023-09-26" @default.
• W2963859359 title "On the enumeration of interval graphs" @default.
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• W2963859359 doi "https://doi.org/10.1090/bproc/27" @default.
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