Cyclic sieving for plane partitions and symmetry. (English) Zbl 1461.05240
Author’s abstract: The cyclic sieving phenomenon of V. Reiner et al. [Notices Am. Math. Soc. 61, No. 2, 169–171 (2014; Zbl 1338.05012)] says that we can often count the fixed points of elements of a cyclic group acting on a combinatorial set by plugging roots of unity into a polynomial related to this set. One of the most impressive instances of the cyclic sieving phenomenon is a theorem of B. Rhoades [J. Comb. Theory, Ser. A 117, No. 1, 38–76 (2010; Zbl 1230.05289)] asserting that the set of plane partitions in a rectangular box under the action of promotion exhibits cyclic sieving. In Rhoades’ result the sieving polynomial is the size generating function for these plane partitions, which has a well-known product formula due to P. A. MacMahon [Combinatory analysis. Vols. I, II (bound in one volume). Reprint of An introduction to combinatory analysis (1920) and Combinatory analysis. Vol. I, II (1915, 1916). Mineola, NY: Dover Publications (2004; Zbl 1144.05300)]. We extend Rhoades’ result by also considering symmetries of plane partitions: specifically, complementation and transposition. The relevant polynomial here is the size generating function for symmetric plane partitions, whose product formula was conjectured by MacMahon [loc. cit.] and proved by G. E. Andrews [in: Studies in foundations and combinatorics. New York, NY, San Francisco, CA, London: Academic Press. 131–150 (1978; Zbl 0462.10010)] and I. G. Macdonald [Symmetric functions and Hall polynomials. Oxford University Press, Oxford (1979; Zbl 0487.20007)]. Finally, we explain how these symmetry results also apply to the rowmotion operator on plane partitions, which is closely related to promotion.
Reviewer: Andrea Svob (Rijeka)
MSC:
| 05E18 | Group actions on combinatorial structures |
| 05E10 | Combinatorial aspects of representation theory |
| 17B10 | Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) |
| 17B37 | Quantum groups (quantized enveloping algebras) and related deformations |
References:
| [1] | Abuzzahab, O. and Korson, M. and Chun, M. L. M. and Meyer, S., Cyclic and dihedral sieving phenomenon for plane partitions |
| [2] | Alexandersson, Per and Linusson, Svante and Potka, Samu, The cyclic sieving phenomenon on circular {D}yck paths, Electronic Journal of Combinatorics, 26, 4, 4.16, 32 pages, (2019) · Zbl 1422.05006 · doi:10.37236/8720 |
| [3] | Andrews, George E., Plane partitions. {II}. {T}he equivalence of the {B}ender–{K}nuth and {M}ac{M}ahon conjectures, Pacific Journal of Mathematics, 72, 2, 283-291, (1977) · Zbl 0376.10014 · doi:10.2140/pjm.1977.72.283 |
| [4] | Andrews, George E., Plane partitions. {I}. {T}he {M}ac{M}ahon conjecture, Studies in Foundations and Combinatorics, Adv. in Math. Suppl. Stud., 1, 131-150, (1978), Academic Press, New York – London · Zbl 0462.10010 |
| [5] | Armstrong, Drew and Stump, Christian and Thomas, Hugh, A uniform bijection between nonnesting and noncrossing partitions, Transactions of the American Mathematical Society, 365, 8, 4121-4151, (2013) · Zbl 1271.05011 · doi:10.1090/S0002-9947-2013-05729-7 |
| [6] | Barcelo, H\'el\`ene and Reiner, Victor and Stanton, Dennis, Bimahonian distributions, Journal of the London Mathematical Society. Second Series, 77, 3, 627-646, (2008) · Zbl 1202.05147 · doi:10.1112/jlms/jdn004 |
| [7] | Bender, Edward A. and Knuth, Donald E., Enumeration of plane partitions, Journal of Combinatorial Theory. Series A, 13, 40-54, (1972) · Zbl 0246.05010 · doi:10.1016/0097-3165(72)90007-6 |
| [8] | Berenstein, Arkady and Zelevinsky, Andrei, Canonical bases for the quantum group of type {\(A_r\)} and piecewise-linear combinatorics, Duke Mathematical Journal, 82, 3, 473-502, (1996) · Zbl 0898.17006 · doi:10.1215/S0012-7094-96-08221-6 |
| [9] | Bloom, Jonathan and Pechenik, Oliver and Saracino, Dan, Proofs and generalizations of a homomesy conjecture of {P}ropp and {R}oby, Discrete Mathematics, 339, 1, 194-206, (2016) · Zbl 1322.05136 · doi:10.1016/j.disc.2015.08.011 |
| [10] | Brouwer, A. E. and Schrijver, A., On the period of an operator, defined on antichains, i+13, (1974), Mathematisch Centrum, Amsterdam · Zbl 0282.06003 |
| [11] | Brundan, Jonathan, Dual canonical bases and {K}azhdan–{L}usztig polynomials, Journal of Algebra, 306, 1, 17-46, (2006) · Zbl 1169.17008 · doi:10.1016/j.jalgebra.2006.01.053 |
| [12] | Cameron, P. J. and Fon-Der-Flaass, D. G., Orbits of antichains revisited, European Journal of Combinatorics, 16, 6, 545-554, (1995) · Zbl 0831.06001 · doi:10.1016/0195-6698(95)90036-5 |
| [13] | Chmutov, Michael and Glick, Max and Pylyavskyy, Pavlo, The {B}erenstein–{K}irillov group and cactus groups, Journal of Combinatorial Algebra, 4, 2, 111-140, (2020) · Zbl 1446.05091 · doi:10.4171/JCA/36 |
| [14] | Du, Jie, Canonical bases for irreducible representations of quantum {\({\rm GL}_n\)}, The Bulletin of the London Mathematical Society, 24, 4, 325-334, (1992) · Zbl 0770.17005 · doi:10.1112/blms/24.4.325 |
| [15] | Du, Jie, Canonical bases for irreducible representations of quantum {\({\rm GL}_n\)}. {II}, Journal of the London Mathematical Society. Second Series, 51, 3, 461-470, (1995) · Zbl 0999.17021 · doi:10.1112/jlms/51.3.461 |
| [16] | Einstein, David and Propp, James, Combinatorial, piecewise-linear, and birational homomesy for products of two chains · Zbl 1465.05194 |
| [17] | Einstein, David and Propp, James, Piecewise-linear and birational toggling, 26th {I}nternational {C}onference on {F}ormal {P}ower {S}eries and {A}lgebraic {C}ombinatorics ({FPSAC} 2014), Discrete Math. Theor. Comput. Sci. Proc., AT, 513-524, (2014), Assoc. Discrete Math. Theor. Comput. Sci., Nancy · Zbl 1394.06005 |
| [18] | Fock, Vladimir V. and Goncharov, Alexander B., Cluster ensembles, quantization and the dilogarithm, Annales Scientifiques de l’\'Ecole Normale Sup\'erieure. Quatri\`eme S\'erie, 42, 6, 865-930, (2009) · Zbl 1180.53081 · doi:10.24033/asens.2112 |
| [19] | Fon-Der-Flaass, D. G., Orbits of antichains in ranked posets, European Journal of Combinatorics, 14, 1, 17-22, (1993) · Zbl 0777.06002 · doi:10.1006/eujc.1993.1003 |
| [20] | Fontaine, Bruce and Kamnitzer, Joel, Cyclic sieving, rotation, and geometric representation theory, Selecta Mathematica. New Series, 20, 2, 609-625, (2014) · Zbl 1295.22018 · doi:10.1007/s00029-013-0144-4 |
| [21] | Fraser, Chris, Braid group symmetries of {G}rassmannian cluster algebras, Selecta Mathematica. New Series, 26, 2, 17, 51 pages, (2020) · Zbl 1436.13049 · doi:10.1007/s00029-020-0542-3 |
| [22] | Frieden, Gabriel, Affine type {\(A\)} geometric crystal on the {G}rassmannian, Journal of Combinatorial Theory. Series A, 167, 499-563, (2019) · Zbl 1467.17016 · doi:10.1016/j.jcta.2019.05.010 |
| [23] | Fuchs, J\"urgen and Ray, Urmie and Schweigert, Christoph, Some automorphisms of generalized {K}ac–{M}oody algebras, Journal of Algebra, 191, 2, 518-540, (1997) · Zbl 0885.17016 · doi:10.1006/jabr.1996.6907 |
| [24] | Fuchs, J\"urgen and Schellekens, Bert and Schweigert, Christoph, From {D}ynkin diagram symmetries to fixed point structures, Communications in Mathematical Physics, 180, 1, 39-97, (1996) · Zbl 0863.17020 · doi:10.1007/BF02101182 |
| [25] | Fulton, William and Harris, Joe, Representation theory: a first course, Graduate Texts in Mathematics, 129, xvi+551, (1991), Springer-Verlag, New York · Zbl 0744.22001 · doi:10.1007/978-1-4612-0979-9 |
| [26] | Galashin, Pavel and Pylyavskyy, Pavlo, {\(R\)}-systems, Selecta Mathematica. New Series, 25, 2, 22, 63 pages, (2019) · Zbl 1460.37041 · doi:10.1007/s00029-019-0470-2 |
| [27] | Gansner, Emden R., On the equality of two plane partition correspondences, Discrete Mathematics, 30, 2, 121-132, (1980) · Zbl 0467.05009 · doi:10.1016/0012-365X(80)90114-4 |
| [28] | Garver, Alexander and Patrias, Rebecca and Thomas, Hugh, Minuscule reverse plane partitions via quiver representations, S\'eminaire Lotharingien de Combinatoire, 82B, 44, 12 pages, (2020) · Zbl 1436.05018 |
| [29] | Gordon, Basil, A proof of the {B}ender–{K}nuth conjecture, Pacific Journal of Mathematics, 108, 1, 99-113, (1983) · Zbl 0533.05005 · doi:10.2140/pjm.1983.108.99 |
| [30] | Grinberg, Darij and Roby, Tom, Iterative properties of birational rowmotion {II}: rectangles and triangles, Electronic Journal of Combinatorics, 22, 3, 3.40, 49 pages, (2015) · Zbl 1339.06001 · doi:10.37236/4335 |
| [31] | Grinberg, Darij and Roby, Tom, Iterative properties of birational rowmotion {I}: generalities and skeletal posets, Electronic Journal of Combinatorics, 23, 1, 1.33, 40 pages, (2016) · Zbl 1338.06003 · doi:10.37236/4334 |
| [32] | Grojnowski, I. and Lusztig, G., A comparison of bases of quantized enveloping algebras, Linear Algebraic Groups and their Representations ({L}os {A}ngeles, {CA}, 1992), Contemp. Math., 153, 11-19, (1993), Amer. Math. Soc., Providence, RI · Zbl 1009.17502 · doi:10.1090/conm/153/01304 |
| [33] | Gross, Mark and Hacking, Paul and Keel, Sean and Kontsevich, Maxim, Canonical bases for cluster algebras, Journal of the American Mathematical Society, 31, 2, 497-608, (2018) · Zbl 1446.13015 · doi:10.1090/jams/890 |
| [34] | Haiman, Mark D., Dual equivalence with applications, including a conjecture of {P}roctor, Discrete Mathematics, 99, 1-3, 79-113, (1992) · Zbl 0760.05093 · doi:10.1016/0012-365X(92)90368-P |
| [35] | Hopkins, Sam, Minuscule doppelg\"angers, the coincidental down-degree expectations property, and rowmotion, Experimental Mathematics · Zbl 07589797 · doi:10.1080/10586458.2020.1731881 |
| [36] | Jantzen, Jens Carsten, Darstellungen halbeinfacher algebraischer {G}ruppen und zugeordnete kontravariante {F}ormen, Bonner Mathematische Schriften, 67, v+124 pages, (1973) · Zbl 0288.17004 |
| [37] | Karp, Steven N., Moment curves and cyclic symmetry for positive {G}rassmannians, Bulletin of the London Mathematical Society, 51, 5, 900-916, (2019) · Zbl 1443.14048 · doi:10.1112/blms.12280 |
| [38] | Karpman, Rachel, Total positivity for the {L}agrangian {G}rassmannian, Advances in Applied Mathematics, 98, 25-76, (2018) · Zbl 1390.05247 · doi:10.1016/j.aam.2018.02.001 |
| [39] | Karpman, Rachel, The purity conjecture in type {C}, Algebraic Combinatorics, 3, 6, 1401-1416, (2020) · Zbl 1455.05081 · doi:10.5802/alco.145 |
| [40] | Karpman, Rachel and Su, Yi, Combinatorics of symmetric plabic graphs, Journal of Combinatorics, 9, 2, 259-278, (2018) · Zbl 1378.05033 · doi:10.4310/JOC.2018.v9.n2.a3 |
| [41] | Kashiwara, Masaki, Global crystal bases of quantum groups, Duke Mathematical Journal, 69, 2, 455-485, (1993) · Zbl 0774.17018 · doi:10.1215/S0012-7094-93-06920-7 |
| [42] | Kirillov, A. N. and Berenstein, A. D., Groups generated by involutions, {G}el’fand–{T}setlin patterns, and combinatorics of {Y}oung tableaux, St. Petersburg Math. J., 7, 1, 77-127, (1996) · Zbl 0848.20007 |
| [43] | Knight, Harold and Zelevinsky, Andrei, Representations of quivers of type {\(A\)} and the multisegment duality, Advances in Mathematics, 117, 2, 273-293, (1996) · Zbl 0915.16009 · doi:10.1006/aima.1996.0013 |
| [44] | Krattenthaler, C., Plane partitions in the work of {R}ichard {S}tanley and his school, The Mathematical Legacy of {R}ichard {P}. {S}tanley, 231-261, (2016), Amer. Math. Soc., Providence, RI · Zbl 1365.05016 · doi:10.1090//mbk/100/14 |
| [45] | Kuperberg, Greg, Self-complementary plane partitions by {P}roctor’s minuscule method, European Journal of Combinatorics, 15, 6, 545-553, (1994) · Zbl 0816.05005 · doi:10.1006/eujc.1994.1056 |
| [46] | Lam, Thomas, Cyclic {D}emazure modules and positroid varieties, Electronic Journal of Combinatorics, 26, 2, 2.28, 20 pages, (2019) · Zbl 1475.05170 · doi:10.37236/8383 |
| [47] | Lascoux, Alain and Leclerc, Bernard and Thibon, Jean-Yves, Green polynomials and {H}all–{L}ittlewood functions at roots of unity, European Journal of Combinatorics, 15, 2, 173-180, (1994) · Zbl 0789.05093 · doi:10.1006/eujc.1994.1019 |
| [48] | Lenart, Cristian, On the combinatorics of crystal graphs. {I}. {L}usztig’s involution, Advances in Mathematics, 211, 1, 204-243, (2007) · Zbl 1129.05058 · doi:10.1016/j.aim.2006.08.002 |
| [49] | Lusztig, George, Canonical bases arising from quantized enveloping algebras, Journal of the American Mathematical Society, 3, 2, 447-498, (1990) · Zbl 0703.17008 · doi:10.2307/1990961 |
| [50] | Lusztig, George, Introduction to quantum groups, Progress in Mathematics, 110, xii+341, (1993), Birkh\"auser Boston, Inc., Boston, MA · Zbl 1246.17018 · doi:10.1007/978-0-8176-4717-9 |
| [51] | Macdonald, I. G., Symmetric functions and {H}all polynomials, Oxford Mathematical Monographs, viii+180, (1979), The Clarendon Press, Oxford University Press, New York · Zbl 0487.20007 |
| [52] | MacMahon, P. A., Partitions of numbers whose graphs possess symmetry, Transactions of the Cambridge Philosophical Society, 17, 1, 149-170, (1899) · JFM 30.0201.04 |
| [53] | MacMahon, Percy A., Combinatory analysis, {V}ols. {I}, {II}, Dover Phoenix Editions, (2004), Dover Publications, Inc., Mineola, NY · Zbl 1144.05300 |
| [54] | Musiker, Gregg and Roby, Tom, Paths to understanding birational rowmotion on products of two chains, Algebraic Combinatorics, 2, 2, 275-304, (2019) · Zbl 1516.06004 · doi:10.5802/alco.43 |
| [55] | Panyushev, Dmitri I., On orbits of antichains of positive roots, European Journal of Combinatorics, 30, 2, 586-594, (2009) · Zbl 1165.06001 · doi:10.1016/j.ejc.2008.03.009 |
| [56] | Postnikov, Alexander, Total positivity, {G}rassmannians, and networks |
| [57] | Proctor, Robert A., Shifted plane partitions of trapezoidal shape, Proceedings of the American Mathematical Society, 89, 3, 553-559, (1983) · Zbl 0525.05007 · doi:10.2307/2045516 |
| [58] | Proctor, Robert A., Bruhat lattices, plane partition generating functions, and minuscule representations, European Journal of Combinatorics, 5, 4, 331-350, (1984) · Zbl 0562.05003 · doi:10.1016/S0195-6698(84)80037-2 |
| [59] | Proctor, Robert A., New symmetric plane partition identities from invariant theory work of {D}e {C}oncini and {P}rocesi, European Journal of Combinatorics, 11, 3, 289-300, (1990) · Zbl 0726.05008 · doi:10.1016/S0195-6698(13)80128-X |
| [60] | Rao, Sujit and Suk, Joe, Dihedral sieving phenomena, Discrete Mathematics, 343, 6, 111849, 12 pages, (2020) · Zbl 1437.05239 · doi:10.1016/j.disc.2020.111849 |
| [61] | Reiner, Victor and Stanton, Dennis and White, Dennis, The cyclic sieving phenomenon, Journal of Combinatorial Theory. Series A, 108, 1, 17-50, (2004) · Zbl 1052.05068 · doi:10.1016/j.jcta.2004.04.009 |
| [62] | Reiner, Victor and Stanton, Dennis and White, Dennis, What is …cyclic sieving?, Notices of the American Mathematical Society, 61, 2, 169-171, (2014) · Zbl 1338.05012 · doi:10.1090/noti1084 |
| [63] | Rhoades, Brendon, Cyclic sieving, promotion, and representation theory, Journal of Combinatorial Theory. Series A, 117, 1, 38-76, (2010) · Zbl 1230.05289 · doi:10.1016/j.jcta.2009.03.017 |
| [64] | Rhoades, Brendon and Skandera, Mark, Kazhdan–{L}usztig immanants and products of matrix minors, Journal of Algebra, 304, 2, 793-811, (2006) · Zbl 1129.15005 · doi:10.1016/j.jalgebra.2005.07.017 |
| [65] | Rhoades, Brendon and Skandera, Mark, Kazhdan–{L}usztig immanants and products of matrix minors. {II}, Linear and Multilinear Algebra, 58, 1-2, 137-150, (2010) · Zbl 1200.15004 · doi:10.1080/03081080701646638 |
| [66] | Rietsch, K. and Williams, L., Newton–{O}kounkov bodies, cluster duality, and mirror symmetry for {G}rassmannians, Duke Mathematical Journal, 168, 18, 3437-3527, (2019) · Zbl 1439.14142 · doi:10.1215/00127094-2019-0028 |
| [67] | Roby, Tom, Dynamical algebraic combinatorics and the homomesy phenomenon, Recent trends in combinatorics, IMA Vol. Math. Appl., 159, 619-652, (2016), Springer, Cham · Zbl 1354.05146 · doi:10.1007/978-3-319-24298-9_25 |
| [68] | Rush, David B., Restriction of global bases and {R}hoades’s theorem · Zbl 1497.17021 |
| [69] | Rush, David B. and Shi, XiaoLin, On orbits of order ideals of minuscule posets, Journal of Algebraic Combinatorics. An International Journal, 37, 3, 545-569, (2013) · Zbl 1283.06007 · doi:10.1007/s10801-012-0380-2 |
| [70] | Sagan, Bruce E., The cyclic sieving phenomenon: a survey, Surveys in Combinatorics 2011, London Math. Soc. Lecture Note Ser., 392, 183-233, (2011), Cambridge University Press, Cambridge · Zbl 1233.05028 |
| [71] | The Sage-Combinat Community, {S}age-{C}ombinat: enhancing {S}age as a toolbox for computer exploration in algebraic combinatorics |
| [72] | The Sage Developers, {S}agemath, the {S}age {M}athematics {S}oftware {S}ystem ({V}ersion 8.4) |
| [73] | Sch\"utzenberger, M. P., Quelques remarques sur une construction de {S}chensted, Mathematica Scandinavica, 12, 117-128, (1963) · Zbl 0216.30202 · doi:10.7146/math.scand.a-10676 |
| [74] | Sch\"utzenberger, M. P., Promotion des morphismes d’ensembles ordonn\'es, Discrete Mathematics, 2, 73-94, (1972) · Zbl 0279.06001 · doi:10.1016/0012-365X(72)90062-3 |
| [75] | Sch\"utzenberger, M. P., La correspondance de {R}obinson, Combinatoire et repr\'esentation du groupe sym\'etrique, Lecture Notes in Math., 579, 59-113, (1977), Springer-Verlag, Berlin – New York · Zbl 0398.05011 · doi:10.1007/BFb0090012 |
| [76] | Scott, Joshua S., Grassmannians and cluster algebras, Proceedings of the London Mathematical Society. Third Series, 92, 2, 345-380, (2006) · Zbl 1088.22009 · doi:10.1112/S0024611505015571 |
| [77] | Seshadri, C. S., Introduction to the theory of standard monomials, Texts and Readings in Mathematics, 46, xiv+221, (2014), Hindustan Book Agency, New Delhi · Zbl 1297.14002 |
| [78] | Sheats, Jeffrey T., A symplectic jeu de taquin bijection between the tableaux of {K}ing and of {D}e {C}oncini, Transactions of the American Mathematical Society, 351, 9, 3569-3607, (1999) · Zbl 0940.05069 · doi:10.1090/S0002-9947-99-02166-2 |
| [79] | Shen, Linhui and Weng, Daping, Cyclic sieving and cluster duality of {G}rassmannian, SIGMA. Symmetry, Integrability and Geometry. Methods and Applications, 16, 067, 41 pages, (2020) · Zbl 1471.13048 · doi:10.3842/SIGMA.2020.067 |
| [80] | Skandera, Mark, On the dual canonical and {K}azhdan–{L}usztig bases and 3412-, 4231-avoiding permutations, Journal of Pure and Applied Algebra, 212, 5, 1086-1104, (2008) · Zbl 1220.05132 · doi:10.1016/j.jpaa.2007.09.007 |
| [81] | Stanley, Richard P., Symmetries of plane partitions, Journal of Combinatorial Theory. Series A, 43, 1, 103-113, (1986) · Zbl 0602.05007 · doi:10.1016/0097-3165(86)90028-2 |
| [82] | Stanley, Richard P., Enumerative combinatorics, {V}ol. 2, Cambridge Studies in Advanced Mathematics, 62, xii+581, (1999), Cambridge University Press, Cambridge · Zbl 0928.05001 · doi:10.1017/CBO9780511609589 |
| [83] | Stanley, Richard P., Promotion and evacuation, Electronic Journal of Combinatorics, 16, 2, Special volume in honor of Anders Bj\"{o}rner, 9, 24 pages, (2009) · Zbl 1169.06002 · doi:10.37236/75 |
| [84] | Stanley, Richard P., Enumerative combinatorics, {V}ol. 1, Cambridge Studies in Advanced Mathematics, 49, xiv+626, (2012), Cambridge University Press, Cambridge · Zbl 1247.05003 · doi:10.1017/CBO9781139058520 |
| [85] | Stembridge, John R., On minuscule representations, plane partitions and involutions in complex {L}ie groups, Duke Mathematical Journal, 73, 2, 469-490, (1994) · Zbl 0805.22006 · doi:10.1215/S0012-7094-94-07320-1 |
| [86] | Stembridge, John R., Some hidden relations involving the ten symmetry classes of plane partitions, Journal of Combinatorial Theory. Series A, 68, 2, 372-409, (1994) · Zbl 0809.05007 · doi:10.1016/0097-3165(94)90112-0 |
| [87] | Stembridge, John R., Canonical bases and self-evacuating tableaux, Duke Mathematical Journal, 82, 3, 585-606, (1996) · Zbl 0869.17011 · doi:10.1215/S0012-7094-96-08224-1 |
| [88] | Stier, Zachary and Wellman, Julian and Xu, Zixuan, Dihedral sieving on cluster complexes · Zbl 1512.05399 |
| [89] | Striker, Jessica and Williams, Nathan, Promotion and rowmotion, European Journal of Combinatorics, 33, 8, 1919-1942, (2012) · Zbl 1260.06004 · doi:10.1016/j.ejc.2012.05.003 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
