Enumeration of even and odd chord diagrams
Diskretnyj analiz i issledovanie operacij, Tome 31 (2024) no. 2, pp. 63-79 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

A general method for the enumeration of various classes of chord diagrams with even and odd numbers of chord intersections is considered. The method is based on the calculation of the Pfaffian and Hafnian of the constraint matrix characterizing a class of diagrams. Tab. 3, illustr. 6, bibliogr. 23.
Keywords: chord diagram, Pfaffian, Hafnian.
@article{DA_2024_31_2_a3,
     author = {D. B. Efimov},
     title = {Enumeration of even and odd chord diagrams},
     journal = {Diskretnyj analiz i issledovanie operacij},
     pages = {63--79},
     year = {2024},
     volume = {31},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DA_2024_31_2_a3/}
}
TY  - JOUR
AU  - D. B. Efimov
TI  - Enumeration of even and odd chord diagrams
JO  - Diskretnyj analiz i issledovanie operacij
PY  - 2024
SP  - 63
EP  - 79
VL  - 31
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/DA_2024_31_2_a3/
LA  - ru
ID  - DA_2024_31_2_a3
ER  - 
%0 Journal Article
%A D. B. Efimov
%T Enumeration of even and odd chord diagrams
%J Diskretnyj analiz i issledovanie operacij
%D 2024
%P 63-79
%V 31
%N 2
%U http://geodesic.mathdoc.fr/item/DA_2024_31_2_a3/
%G ru
%F DA_2024_31_2_a3
D. B. Efimov. Enumeration of even and odd chord diagrams. Diskretnyj analiz i issledovanie operacij, Tome 31 (2024) no. 2, pp. 63-79. http://geodesic.mathdoc.fr/item/DA_2024_31_2_a3/

[1] Acan H., An enumerative-probabilistic study of chord diagrams, PhD Thes., Ohio State Univ., Columbus, 2013, 144 pp. | MR

[2] E. S. Krasko, I. N. Labutin, and A. V. Omelchenko, “Enumeration of labelled and unlabelled Hamiltonian cycles in complete $k$-partite graphs”, Zap. Nauchn. Semin. POMI, 488, 2019, 119–142 (Russian)

[3] Nakamigawa T., “The expansion of a chord diagram and the Genocchi numbers”, Ars Math. Contemp., 18 (2020), 381–391 | DOI | MR | Zbl

[4] Sullivan E., “Linear chord diagrams with long chords”, Electron. J. Comb., 24:4 (2017), P4.20, 8 pp. | MR

[5] Cameron N. T., Killpatrick K., “Statistics on linear chord diagrams”, Discrete Math. Theor. Comput. Sci., 21:2 (2020), 11, 10 pp. | MR

[6] Acan H., “On a uniformly random chord diagram and its intersection graph”, Discrete Math., 340:8 (2017), 1967–1985 | DOI | MR | Zbl

[7] Touchard J., “Sur un problème de configurations et sur les fractions continues”, Canad. J. Math., 4 (1952), 2–25 (French) | DOI | MR | Zbl

[8] Riordan J., “The distribution of crossing of chords joining pairs of $2n$ points on a circle”, Math. Comput., 29:129 (1975), 215–222 | MR | Zbl

[9] Courtiel J., Yeats K., Zeilberger N., “Connected chord diagrams and bridgeless maps”, Electron. J. Comb., 26:4 (2019), P4.37, 56 pp. | MR | Zbl

[10] Mahmoud A. A., Yeats K., “Connected chord diagrams and the combinatorics of asymptotic expansions”, J. Integer Seq., 25:7 (2022), 22.7.5, 22 pp. | MR | Zbl

[11] Shevelev V. S., “Combinatorial minors for matrix functions and their applications”, Zesz. Nauk. Politech. Śląsk. Ser. Mat. Stosow, 2014, no. 4, 5–16 | MR

[12] V. S. Shevelev, “Some problems of the theory of enumerating the permutations with restricted positions”, J. Sov. Math., 61:4 (1992), 2272–2317 | DOI | MR | Zbl

[13] H. Minc, Permanents, Addison-Wesley, Reading, MA, 1978 | MR | MR | Zbl

[14] Stembridge J. R., “Nonintersecting paths, Pfaffians, and plane partitions”, Adv. Math., 83 (1990), 96–131 | DOI | MR

[15] Barvinok A., Combinatorics and complexity of partition functions, Algorithms Comb., 30, Springer, Cham, 2016, 304 pp. | MR | Zbl

[16] Fishel S., Grojnowski I., “Canonical bases for the Brauer centralizer algebra”, Math. Res. Lett., 2 (1995), 15–26 | DOI | MR | Zbl

[17] Barcelo H., Ram A., “Combinatorial representation theory”, New perspectives in algebraic combinatorics, Math. Sci. Res. Inst. Publ., 38, Camb. Univ. Press, New York, 1999, 23–90 | MR | Zbl

[18] Shalile A., “On the center of the Brauer algebra”, Algebr. Represent. Theory, 16 (2013), 65–100 | DOI | MR | Zbl

[19] Schwartz M., “Efficiently computing the permanent and Hafnian of some banded Toeplitz matrices”, Linear Algebra Appl., 430 (2009), 1364–1374 | DOI | MR | Zbl

[20] L. Lovász and M. D. Plummer, Matching Theory, North-Holland, Amsterdam, 1986 | MR | Zbl

[21] Gabiati G., Maffioli F., “On the computation of Pfaffians”, Discrete Appl. Math., 51 (1994), 269–275 | DOI | MR

[22] Wimmer M., “Algorithm 923: Efficient numerical computation of the Pfaffian for dense and banded skew-symmetric matrices”, ACM Trans. Math. Softw., 38:4 (2012), 30, 17 pp. | DOI | MR | Zbl

[23] The on-line encyclopedia of integer sequences, , OEIS Found., Highland Park, NJ, 2024 (accessed Mar. 3, 2024) oeis.org