A277869 -id:A277869

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A354588

Number of marked chord diagrams (linear words in which each letter appears twice) with n chords, whose intersection graph is connected and distance-hereditary.

+10
1

1, 4, 27, 226, 2116, 21218, 222851, 2420134, 26954622, 306203536, 3534170486, 41326973520, 488562349730, 5829471835390, 70112478797987, 849110215237094, 10345827793291654, 126734013316914248, 1559884942820510474, 19281814963272771308, 239263099541276559360, 2979328903819471935332

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

For n < 5, all intersection graphs on n vertices are distance-hereditary, so the first 4 terms coincide with the number of linear chord diagrams with connected intersection graph.

LINKS

Table of n, a(n) for n=0..21.

Christopher-Lloyd Simon, Topologie et dénombrement des courbes algébriques réelles, arXiv:2106.15450 [math.AG], 2021.

Christopher-Lloyd Simon, Topologie et dénombrement des courbes algébriques réelles, Annales de la faculté des sciences de Toulouse : Mathématiques, 6e série, 31(2): 383--422, 2022.

FORMULA

a(n) = (1/(n+1))*Sum_{k=0..n} binomial(n+k, n)*binomial(2*(n+1)+k, n-k)*2^k.

G.f. satisfies C = z + 2*z*C + (z+2)*C^2 + 2*C^3.

PROG

(PARI) a(n) = sum(k=0, n, binomial(n+k, n)*binomial(2*(n+1)+k, n-k)*2^k)/(n+1); \\ Michel Marcus, Oct 05 2022

CROSSREFS

Cf. A277862, A277869, A357596.

KEYWORD

nonn

AUTHOR

Christopher-Lloyd Simon, May 31 2022

STATUS

approved

A357596

Number of marked chord diagrams (linear words in which each letter appears twice) with n chords, whose intersection graph is distance-hereditary.

+10
1

1, 1, 3, 15, 105, 923, 9417, 105815, 1267681, 15875631, 205301361

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

For n < 5, all intersection graphs on n vertices are distance-hereditary, so the first 4 terms coincide with the number of linear chord diagrams (given by the double factorial numbers; see A001147).

LINKS

Table of n, a(n) for n=0..10.

Christopher-Lloyd Simon, Topologie et dénombrement des courbes algébriques réelles, arXiv:2106.15450 [math.AG], 2021.

Christopher-Lloyd Simon, Topologie et dénombrement des courbes algébriques réelles, Annales de la faculté des sciences de Toulouse : Mathématiques, 6e série, 31(2): 383--422, 2022.

FORMULA

G.f. is algebraic: (z^3 + z^2)*A^6 - z^2*A^5 - 4*z*A^4 + (8*z + 2)*A^3 - (4*z + 6)*A^2 + 6*A - 2 = 0.

CROSSREFS

Cf. A277862, A277869, A354588.

KEYWORD

nonn,more

AUTHOR

Christopher-Lloyd Simon, Oct 05 2022

STATUS

approved

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