Linear chord diagrams on two intervals

Jørgen Ellegaard Andersen, Robert Penner, Christian Reidys, Rita Wang

Publikation: Working paperForskning

Resumé

Consider all possible ways of attaching disjoint chords to two ordered and oriented disjoint intervals so as to produce a connected graph. Taking the intervals to lie in the real axis with the induced orientation and the chords to lie in the upper half plane canonically determines a corresponding fatgraph which has some associated genus $g\geq 0$, and we consider the natural generating function ${\bf C}_g^{[2]}(z)=\sum_{n\geq 0} {\bf c}^{[2]}_g(n)z^n$ for the number ${\bf c}^{[2]}_g(n)$ of distinct such chord diagrams of fixed genus $g\geq 0$ with a given number $n\geq 0$ of chords. We prove here the surprising fact that ${\bf C}^{[2]}_g(z)=z^{2g+1} R_g^{[2]}(z)/(1-4z)^{3g+2} $ is a rational function, for $g\geq 0$, where the polynomial $R^{[2]}_g(z)$ with degree at most $g$ has integer coefficients and satisfies $R_g^{[2]}({1\over 4})\neq 0$. Earlier work had already determined that the analogous generating function ${\bf C}_g(z)=z^{2g}R_g(z)/(1-4z)^{3g-{1\over 2}}$ for chords attached to a single interval is algebraic, for $g\geq 1$, where the polynomial $R_g(z)$ with degree at most $g-1$ has integer coefficients and satisfies $R_g(1/4)\neq 0$ in analogy to the generating function ${\bf C}_0(z)$ for the Catalan numbers. The new results here on ${\bf C}_g^{[2]}(z)$ rely on this earlier work, and indeed, we find that $R_g^{[2]}(z)=R_{g+1}(z) -z\sum_{g_1=1}^g R_{g_1}(z) R_{g+1-g_1}(z)$, for $g\geq 1$.
OriginalsprogEngelsk
UdgiverarXiv.org
StatusUdgivet - 2010
Udgivet eksterntJa

Fingeraftryk

Chord Diagrams
Chord or secant line
Generating Function
Interval
Genus
Disjoint
Catalan number
Polynomial
Integer
Coefficient
Half-plane
Rational function
Analogy
Connected graph
Distinct

Citer dette

Andersen, J. E., Penner, R., Reidys, C., & Wang, R. (2010). Linear chord diagrams on two intervals. arXiv.org.
Andersen, Jørgen Ellegaard ; Penner, Robert ; Reidys, Christian ; Wang, Rita. / Linear chord diagrams on two intervals. arXiv.org, 2010.
@techreport{4ced28569226499aa90bcad5e5e03cd8,
title = "Linear chord diagrams on two intervals",
abstract = "Consider all possible ways of attaching disjoint chords to two ordered and oriented disjoint intervals so as to produce a connected graph. Taking the intervals to lie in the real axis with the induced orientation and the chords to lie in the upper half plane canonically determines a corresponding fatgraph which has some associated genus $g\geq 0$, and we consider the natural generating function ${\bf C}_g^{[2]}(z)=\sum_{n\geq 0} {\bf c}^{[2]}_g(n)z^n$ for the number ${\bf c}^{[2]}_g(n)$ of distinct such chord diagrams of fixed genus $g\geq 0$ with a given number $n\geq 0$ of chords. We prove here the surprising fact that ${\bf C}^{[2]}_g(z)=z^{2g+1} R_g^{[2]}(z)/(1-4z)^{3g+2} $ is a rational function, for $g\geq 0$, where the polynomial $R^{[2]}_g(z)$ with degree at most $g$ has integer coefficients and satisfies $R_g^{[2]}({1\over 4})\neq 0$. Earlier work had already determined that the analogous generating function ${\bf C}_g(z)=z^{2g}R_g(z)/(1-4z)^{3g-{1\over 2}}$ for chords attached to a single interval is algebraic, for $g\geq 1$, where the polynomial $R_g(z)$ with degree at most $g-1$ has integer coefficients and satisfies $R_g(1/4)\neq 0$ in analogy to the generating function ${\bf C}_0(z)$ for the Catalan numbers. The new results here on ${\bf C}_g^{[2]}(z)$ rely on this earlier work, and indeed, we find that $R_g^{[2]}(z)=R_{g+1}(z) -z\sum_{g_1=1}^g R_{g_1}(z) R_{g+1-g_1}(z)$, for $g\geq 1$.",
author = "Andersen, {J{\o}rgen Ellegaard} and Robert Penner and Christian Reidys and Rita Wang",
year = "2010",
language = "English",
publisher = "arXiv.org",
type = "WorkingPaper",
institution = "arXiv.org",

}

Andersen, JE, Penner, R, Reidys, C & Wang, R 2010 'Linear chord diagrams on two intervals' arXiv.org.

Linear chord diagrams on two intervals. / Andersen, Jørgen Ellegaard; Penner, Robert; Reidys, Christian; Wang, Rita.

arXiv.org, 2010.

Publikation: Working paperForskning

TY - UNPB

T1 - Linear chord diagrams on two intervals

AU - Andersen, Jørgen Ellegaard

AU - Penner, Robert

AU - Reidys, Christian

AU - Wang, Rita

PY - 2010

Y1 - 2010

N2 - Consider all possible ways of attaching disjoint chords to two ordered and oriented disjoint intervals so as to produce a connected graph. Taking the intervals to lie in the real axis with the induced orientation and the chords to lie in the upper half plane canonically determines a corresponding fatgraph which has some associated genus $g\geq 0$, and we consider the natural generating function ${\bf C}_g^{[2]}(z)=\sum_{n\geq 0} {\bf c}^{[2]}_g(n)z^n$ for the number ${\bf c}^{[2]}_g(n)$ of distinct such chord diagrams of fixed genus $g\geq 0$ with a given number $n\geq 0$ of chords. We prove here the surprising fact that ${\bf C}^{[2]}_g(z)=z^{2g+1} R_g^{[2]}(z)/(1-4z)^{3g+2} $ is a rational function, for $g\geq 0$, where the polynomial $R^{[2]}_g(z)$ with degree at most $g$ has integer coefficients and satisfies $R_g^{[2]}({1\over 4})\neq 0$. Earlier work had already determined that the analogous generating function ${\bf C}_g(z)=z^{2g}R_g(z)/(1-4z)^{3g-{1\over 2}}$ for chords attached to a single interval is algebraic, for $g\geq 1$, where the polynomial $R_g(z)$ with degree at most $g-1$ has integer coefficients and satisfies $R_g(1/4)\neq 0$ in analogy to the generating function ${\bf C}_0(z)$ for the Catalan numbers. The new results here on ${\bf C}_g^{[2]}(z)$ rely on this earlier work, and indeed, we find that $R_g^{[2]}(z)=R_{g+1}(z) -z\sum_{g_1=1}^g R_{g_1}(z) R_{g+1-g_1}(z)$, for $g\geq 1$.

AB - Consider all possible ways of attaching disjoint chords to two ordered and oriented disjoint intervals so as to produce a connected graph. Taking the intervals to lie in the real axis with the induced orientation and the chords to lie in the upper half plane canonically determines a corresponding fatgraph which has some associated genus $g\geq 0$, and we consider the natural generating function ${\bf C}_g^{[2]}(z)=\sum_{n\geq 0} {\bf c}^{[2]}_g(n)z^n$ for the number ${\bf c}^{[2]}_g(n)$ of distinct such chord diagrams of fixed genus $g\geq 0$ with a given number $n\geq 0$ of chords. We prove here the surprising fact that ${\bf C}^{[2]}_g(z)=z^{2g+1} R_g^{[2]}(z)/(1-4z)^{3g+2} $ is a rational function, for $g\geq 0$, where the polynomial $R^{[2]}_g(z)$ with degree at most $g$ has integer coefficients and satisfies $R_g^{[2]}({1\over 4})\neq 0$. Earlier work had already determined that the analogous generating function ${\bf C}_g(z)=z^{2g}R_g(z)/(1-4z)^{3g-{1\over 2}}$ for chords attached to a single interval is algebraic, for $g\geq 1$, where the polynomial $R_g(z)$ with degree at most $g-1$ has integer coefficients and satisfies $R_g(1/4)\neq 0$ in analogy to the generating function ${\bf C}_0(z)$ for the Catalan numbers. The new results here on ${\bf C}_g^{[2]}(z)$ rely on this earlier work, and indeed, we find that $R_g^{[2]}(z)=R_{g+1}(z) -z\sum_{g_1=1}^g R_{g_1}(z) R_{g+1-g_1}(z)$, for $g\geq 1$.

M3 - Working paper

BT - Linear chord diagrams on two intervals

PB - arXiv.org

ER -

Andersen JE, Penner R, Reidys C, Wang R. Linear chord diagrams on two intervals. arXiv.org. 2010.