On Whitney duals of operadic posets Yeison A. Quiceno Dur´an Universidad Nacional de Colombia sede Medell´ın Escuela de Matem´aticas Medell´ın,Colombia 2020 On Whitney duals of operadic posets Yeison A. Quiceno Dur´an Tesis presentada como requisito parcial para optar al t´ıtulode: Magister en Matem´aticas Director: Rafael S. Gonz´alezD’Le´on,Ph.D. Codirector: Juan Diego V´elezCaicedo, Ph.D. L´ıneade Investigaci´on: Combinatoria Algebraica y Topol´ogica Universidad Nacional de Colombia sede Medell´ın Escuela de Matem´aticas Medell´ın,Colombia 2020 Acknowledgements I want to thank my supervisor Rafael S. Gonz´alezD’Le´onfor his time and dedication. He has enriched me with his knowledge in the area of combinatorics and his kind and hum- ble character has made the time spent on this project a very pleasant task. His unique and amazing traits as a scientist have constantly fueled my desire to learn. I can genuinely say this project would have not been possible without his guidance. vii Abstract The notion of a Whitney dual for a graded partially ordered set (poset) P with a mini- mum element 0^ has been introduced recently by Gonz´alezD’Le´onand Hallam with some interesting connections to other areas of algebra and combinatorics. We say that two posets are Whitney duals to each other if (the absolute value of) their Whitney numbers of the first and second kind are interchanged between the two posets. Some families of familiar posets such as the poset Πn of partitions of the set f1; 2; 3:::; ng have Whitney duals. This has been proved by defining a suitable edge labeling λ on the edges of the Hasse diagram of Πn satisfying certain conditions. Such an edge labeling is called a Whitney labeling and Gonz´alezD’Le´on- Hallam proved that every graded poset that admits a Whitney labeling has a Whitney dual. We study the Whitney duality property for two families of operadic posets, finding Whitney labelings and constructing combinatorial descriptions of their Whitney duals. One is known k as the family of posets of weighted partitions Πn, studied by Gonz´alezD’Le´onand Wachs related to the operad Comk of commutative algebras with k totally commutative products, • and the other is the family of posets of pointed partitions Πn, studied by Chapoton and Vallette associated to the operad Perm of Perm-algebras. We prove that a labeling, previ- k ously defined by Gonz´alezD’Le´on,for Πn is a Whitney labeling and prove that its associated • Whitney dual is a poset of colored Lyndon forests. We also find a Whitney labeling for Πn and then use this labeling to show that its associated Whitney dual is a poset of pointed 2 • Lyndon forests. For the case k = 2, it turns out that the families Πn and Πn have the same Whitney numbers of the first and second kind. Our results imply that there are multiple non-isomorphic Whitney duals for these two families in this case. ix Resumen T´ıtulo:Duales de Whitney de posets oper´adicos Gonz´alezD’Le´ony Hallam introdujeron recientemente la noci´onde duales de Whitney para un conjunto parcialmente ordenado (poset) graduado P con un elemento m´ınimo 0^ con al- gunas conexiones interesantes a otras ´areasdel ´algebray la combinatoria. Decimos que dos posets son duales de Whitney entre s´ı,si (el valor absoluto de) sus n´umerosde Whitney del primer y segundo tipo se intercambian entre los dos posets. Algunas familias de posets familiares como el poset Πn de particiones del conjunto f1; 2; 3:::; ng tienen duales de Whit- ney. Esto se ha demostrado definiendo un etiquetamiento adecuado λ en las aristas del diagrama de Hasse de Πn que satisface ciertas condiciones. A tal etiquetamiento de aristas se le llama etiquetamiento de Whitney y Gonz´alezD’Le´on- Hallam demostraron que todo poset graduado que admite un etiquetamiento de Whitney tiene un dual de Whitney. Estudiamos la propiedad de dualidad de Whitney para dos familias de posets operadicos, por medio de etiquetamientos de Whitney y de la construcci´onde descripciones combinatorias de sus duales de Whitney. Una de las familias es la familia de posets de particiones con k k pesos Πn, estudiadas por Gonz´alezD’Le´ony Wachs, relacionadas con el operad Com de ´algebrasconmutativas con k productos totalmente conmutativos, y la otra es la familia de • posets de particiones punteadas Πn, estudiadas por Chapoton y Vallette asociadas al operad Perm de Perm-´algebras. Demostramos que un etiquetamiento, previamente definido por k Gonz´alezD’Le´on,para Πn es un etiquetamiento de Whitney y demostramos que su dual de Whitney asociado es un poset de bosques de Lyndon coloreados. Tambi´enencontramos un • etiquetamiento de Whitney para Πn y luego usamos este etiquetamiento para mostrar que su dual de Whitney asociado es un poset de bosques de Lyndon punteados. Para el caso k = 2, 2 • resulta que las familias Πn y Πn tienen los mismos n´umerosde Whitney del primer y segundo tipo. Nuestros resultados implican que hay m´ultiplesduales de Whitney no isom´orfosentre s´ıpara estas dos familias en este caso. Contents Acknowledgementsv Abstract vii Resumen ix List of Figures1 1 Introduction2 1.1 Whitney duals of a graded poset . .2 1.2 A longstanding conjecture . .3 1.3 Partition posets associated to operads . .4 1.4 Results . .4 1.5 Organization of this thesis . .5 2 Preliminaries6 2.1 Partially ordered sets . .6 2.2 The M¨obiusfunction and the Whitney numbers of the first and second kind7 2.3 Whitney duals and Whitney labelings . .9 2.4 The construction of the Whitney dual Qλ(P ).................. 11 2.5 Operadic partition posets . 12 3 Whitney duals of some operadic posets 15 3.1 The poset of weighted partitions . 15 3.1.1 A Whitney labeling for the poset of weighted partitions . 17 3.1.2 A poset of colored Lyndon forests . 19 3.2 The poset of pointed partitions . 29 3.2.1 The poset of pointed partitions . 29 3.2.2 Poset of pointed Lyndon trees . 34 3.3 Whitney twins: a discussion . 43 4 Open questions and future work 47 4.1 On the necessity of Whitney labelings . 47 4.2 On the uniqueness of Whitney duals and other examples . 48 xii Contents Bibliography 49 List of Figures 2-1 Π3 and its Whitney numbers. .8 2-2 ISF3 and its Whitney numbers. .8 2-3 Examples of posets with or without Whitney duals. .9 2-4 Example of edge labelings on Π3 and ISF3.................... 11 2-5 Example of the construction of Qλ(Π3)...................... 13 2 3-1 Π3.......................................... 16 3-2 SF3 (the roots are represented by squares). 16 2 3-3 QλE (Π3)....................................... 17 3 3-4 Poset of labels Λ3................................. 17 k 3-5 Rank two intervals in Πn............................. 19 3-6 Colored Binary Tree T 2 BT (3;3;2)........................ 20 3-7 Lyn3;2........................................ 21 (3;3;2) 3-8 c(T ) 2 M[0^; [9] ] for T 2 BT (3;3;2) (internal nodes ordered with the reverse-minimal linear extension). 22 3-9 Example of c(F ) for F 2 FLyn8;2......................... 23 3-10 Example of 2-merging two colored Lyndon trees. 25 3-11 FLyn3;2....................................... 28 • 3-12 Π3 with its edge labeling λP ............................ 30 • 3-13 Poset of labels Λ4................................. 31 • 3-14 Rank two intervals in Π3.............................. 35 • 3-15 QλP (Π3)....................................... 36 3-16 Lyn3;•........................................ 36 3-17 c(F ) for F 2 FLyn8;•............................... 39 0 3-18 F 2 FLyn8;• obtained by a 1−merge of T1 and T2, both pointed Lyndon trees in F ...................................... 40 3-19 FLyn3;•....................................... 42 3-20 An interval of rank 3 in FLyn4;2......................... 44 3-21 Rank 3 interval used in the proof of Theorem 3.3.4. 45 3-22 Pointed Lyndon forest in the proof of Theorem 3.3.5. 46 4-1 A Whitney dualizable poset without Whitney labeling. 47 4-2 Example of a Whitney labelable poset that is a Whitney twin of the poset of Figure 4-1..................................... 48 1 Introduction A partially ordered set or poset is a pair (P; ≤) where P is a set and ≤ is a relation that is reflexive, antisymmetric and transitive. In general, we will abuse notation and say that P is a poset. We denote x < y whenever x; y 2 P are such that x ≤ y but x 6= y. If x < y and there is no z 2 P such that x < z < y, we say that y covers x and we denote it by x l y. A subset C of P is said to be a chain if for every pair of elements x; y 2 C either x ≤ y or y ≤ x (that is, x and y are comparable). We say that an element x 2 P is minimal if there is no z 2 P such that z < x. If there is a unique minimal element we call it the minimum of P and we denote it by 0.^ A maximal chain of P is a chain C in P such that for every z 2 P n C the subset C [ fzg is not a chain. A poset whose maximal chains are all of the same cardinality is said to be graded. It is known that in a graded poset we can define a function ρ : P ! N such that ρ(x) = 0 when x is a minimal element and ρ(y) = ρ(x) + 1 whenever x l y. We call ρ the rank function of P .