Browse issues
no. 3
On the association scheme of perfect matchings and their designs
Bamberg, John1; Klawuhn, Lukas2
1The University of Western Australia, Department of Mathematics and Statistics, 35 Stirling Highway, Perth, W.A. 6009 (Australia)
2Paderborn University, Department of Mathematics, Warburger Str. 100, 33098 Paderborn (Germany)
Algebraic Combinatorics, Volume 9 (2026) no. 3, pp. 789-809

We investigate generalisations of 1-factorisations and hyperfactorisations of the complete graph $K_{2n}$. We show that they are special subsets of the association scheme obtained from the Gelfand pair $(S_{2n},S_2 \wr S_n)$. This unifies and extends results by Cameron (1976) and gives rise to new existence and non-existence results. Our methods involve working in the group algebra $\mathbb{C}[S_{2n}]$ and using the representation theory of $S_{2n}$.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.490
Classification: 05E30, 05C70, 05B99
Keywords: perfect matchings, association schemes, Gelfand pairs, zonal spherical functions

Bamberg, John  1 ; Klawuhn, Lukas  2

1 The University of Western Australia, Department of Mathematics and Statistics, 35 Stirling Highway, Perth, W.A. 6009 (Australia)
2 Paderborn University, Department of Mathematics, Warburger Str. 100, 33098 Paderborn (Germany)
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
Bamberg, John; Klawuhn, Lukas. On the association scheme of perfect matchings and their designs. Algebraic Combinatorics, Volume 9 (2026) no. 3, pp. 789-809. doi: 10.5802/alco.490
@article{ALCO_2026__9_3_789_0,
     author = {Bamberg, John and Klawuhn, Lukas},
     title = {On the association scheme of perfect matchings and their designs},
     journal = {Algebraic Combinatorics},
     pages = {789--809},
     year = {2026},
     publisher = {The Combinatorics Consortium},
     volume = {9},
     number = {3},
     doi = {10.5802/alco.490},
     language = {en},
     url = {https://alco.centre-mersenne.org/articles/10.5802/alco.490/}
}
TY  - JOUR
AU  - Bamberg, John
AU  - Klawuhn, Lukas
TI  - On the association scheme of perfect matchings and their designs
JO  - Algebraic Combinatorics
PY  - 2026
SP  - 789
EP  - 809
VL  - 9
IS  - 3
PB  - The Combinatorics Consortium
UR  - https://alco.centre-mersenne.org/articles/10.5802/alco.490/
DO  - 10.5802/alco.490
LA  - en
ID  - ALCO_2026__9_3_789_0
ER  - 
%0 Journal Article
%A Bamberg, John
%A Klawuhn, Lukas
%T On the association scheme of perfect matchings and their designs
%J Algebraic Combinatorics
%D 2026
%P 789-809
%V 9
%N 3
%I The Combinatorics Consortium
%U https://alco.centre-mersenne.org/articles/10.5802/alco.490/
%R 10.5802/alco.490
%G en
%F ALCO_2026__9_3_789_0

[1] Bamberg, John; Lansdown, Jesse Implications of vanishing Krein parameters on Delsarte designs, with applications in finite geometry, Algebr. Comb., Volume 6 (2023) no. 1, pp. 197-212 | Numdam | Zbl | DOI | MR

[2] Bannai, Eiichi; Ito, Tatsuro Algebraic combinatorics. I: Association schemes, The Benjamin/Cummings Publishing Co., Inc., Menlo Park, CA, 1984, xxiv+425 pages | Zbl | MR

[3] Boros, Endre; Jungnickel, Dieter; Vanstone, Scott A. The existence of nontrivial hyperfactorizations of K 2n , Combinatorica, Volume 11 (1991) no. 1, pp. 9-15 | Zbl | DOI | MR

[4] Brouwer, Andries The t-designs with v<18 (1977) no. ZN 76/77 (Technical report)

[5] Cameron, Peter J. Parallelisms of complete designs, London Mathematical Society Lecture Note Series, No. 23, Cambridge University Press, Cambridge-New York-Melbourne, 1976, 144 pages | Zbl | DOI | MR

[6] Cooper, Benjamin C. Abstract Hyperovals, Partial Geometries, and Transitive Hyperovals, Ph. D. Thesis, Colorado State University (2015)

[7] De Clerck, F. The pseudogeometric and geometric (t,s,s-1)-graphs, Simon Stevin, Volume 53 (1979) no. 4, pp. 301-317 | Zbl | MR

[8] Delsarte, P. An algebraic approach to the association schemes of coding theory, Philips Res. Rep. Suppl. (1973) no. 10, p. vi+97 | Zbl | MR

[9] Delsarte, P. Pairs of vectors in the space of an association scheme, Philips Res. Rep., Volume 32 (1977) no. 5-6, pp. 373-411 | MR

[10] Faina, Giorgio The B-ovals of order q≤8, J. Combin. Theory Ser. A, Volume 36 (1984) no. 3, pp. 307-314 | Zbl | DOI | MR

[11] Godsil, Chris; Meagher, Karen Erdős-Ko-Rado theorems: algebraic approaches, Cambridge Studies in Advanced Mathematics, 149, Cambridge University Press, Cambridge, 2016, xvi+335 pages | Zbl | DOI | MR

[12] Gurobi Optimization Inc. Gurobi Optimizer Version 8.1, http://www.gurobi.com/

[13] Jungnickel, Dieter; Vanstone, Scott A. Hyperfactorizations of graphs and 5-designs, J. Univ. Kuwait Sci., Volume 14 (1987) no. 2, pp. 213-224 | Zbl | MR

[14] Lam, C. W. H.; Thiel, L.; Swiercz, S.; McKay, J. The nonexistence of ovals in a projective plane of order 10, Discrete Math., Volume 45 (1983) no. 2-3, pp. 319-321 | Zbl | DOI | MR

[15] Lansdown, Jesse Gurobify - A GAP interface to Gurobi Optimizer, Version 2.0.0, 2021 | DOI

[16] Lindzey, Nathan Intersecting families of perfect matchings, 2018 | arXiv

[17] Macdonald, I. G. Symmetric functions and Hall polynomials, Oxford Classic Texts in the Physical Sciences, The Clarendon Press, Oxford University Press, New York, 2015, xii+475 pages | Zbl | MR

[18] Mathon, Rudolf The partial geometries pg (5,7,3), Congr. Numer., Volume 31 (1981), pp. 129-139 | Zbl | MR

[19] Prince, Alan R. Oval configurations of involutions in symmetric groups, Discrete Math., Volume 174 (1997) no. 1-3, pp. 277-282 Combinatorics (Rome and Montesilvano, 1994) | Zbl | DOI | MR

[20] Rands, B. M. I. An association scheme for the 1-factors of the complete graph, J. Combin. Theory Ser. A, Volume 34 (1983) no. 3, pp. 301-312 | MR | Zbl | DOI

[21] Sagan, Bruce E. The symmetric group: Representations, combinatorial algorithms, and symmetric functions, Graduate Texts in Mathematics, 203, Springer-Verlag, New York, 2001, xvi+238 pages | Zbl | DOI | MR

[22] Serre, Jean-Pierre Linear representations of finite groups, Graduate Texts in Mathematics, Vol. 42, Springer-Verlag, New York-Heidelberg, 1977, x+170 pages | DOI | Zbl | MR

[23] Stinson, D. R. Designs constructed from maximal arcs, Discrete Math., Volume 97 (1991) no. 1-3, pp. 387-393 | Zbl | DOI | MR

[24] Thompson, John G. Fixed point free involutions and finite projective planes, Finite simple groups II, Proc. Lond. Math. Soc. Res. Symp., Univ. Durham 1978, 321-335 (1980)., 1980 | Zbl

Cited by Sources: