Strict Partitions Research Articles

Plane Partitions and a Problem of Josephus

The Josephus Problem is a mathematical counting-out problem with a grim description: given a group of n persons arranged in a circle under the edict that every kth person will be executed going around the circle until only one remains, find the position L(n,k) in which you should stand in order to be the last survivor. Let Jn be the order in which the first person is executed on counting when k=2. In this paper, we consider the sequence (Jn)n⩾1 in order to introduce new expressions for the generating functions of the number of strict plane partitions and the number of symmetric plane partitions. This approach allows us to express the number of strict plane partitions of n and the number of symmetric plane partitions of n as sums over partitions of n in terms of binomial coefficients involving Jn. Also, we introduce interpretations for the strict plane partitions and the symmetric plane partitions in terms of colored partitions. Connections between the sum of the divisors’ functions and Jn are provided in this context.

Beck-type companion identities for Franklin's identity

In 2017, Beck conjectured that the difference in the number of parts in all partitions of $n$ into odd parts and the number of parts in all strict partitions of $n$ is equal to the number of partitions of $n$ whose set of even parts has one element, and also to the number of partitions of $n$ with exactly one part repeated. Andrews proved the conjecture using generating functions. Beck's identity is a companion identity to Euler's identity. The theorem has been generalized (with a combinatorial proof) by Yang to a companion identity to Glaisher's identity. Franklin generalized Glaisher's identity, and in this article, we provide a Beck-type companion identity to Franklin's identity and prove it both analytically and combinatorially. Andrews' and Yang's respective theorems fit naturally into this very general frame. We also discuss how Franklin's identity and the companion Beck-type identities can be further generalized to Euler pairs of any order.

A Tableau Formula for Vexillary Schubert Polynomials in Type C

Ikeda-Mihalcea-Naruse's double Schubert polynomials [Adv. Math. 226 (2011)] represent the equivariant cohomology classes of Schubert varieties in the type C flag varieties. The goal of this paper is to obtain a new tableau formula of these polynomials associated to vexillary signed permutations introduced by Anderson-Fulton. To achieve that goal, we introduce flagged factorial (Schur) $Q$-functions, combinatorially defined functions in terms of marked shifted tableaux for flagged strict partitions, and prove their Schur-Pfaffian formula. As an application, we also obtain a new combinatorial formula of factorial $Q$-functions of Ivanov in which monomials bijectively corresponds to flagged marked shifted tableaux.

Geometry of solutions to the c-projective metrizability equation

On an almost complex manifold, a quasi-Kähler metric, with canonical connection in the c-projective class of a given minimal complex connection, is equivalent to a nondegenerate solution of the c-projectively invariant metrizability equation. For this overdetermined equation, replacing this maximal rank condition on solutions with a nondegeneracy condition on the prolonged system yields a strictly wider class of solutions with non-vanishing (generalized) scalar curvature. We study the geometries induced by this class of solutions. For each solution, the strict point-wise signature partitions the underlying manifold into strata, in a manner that generalizes the model, a certain Lie group orbit decomposition of \(\mathbb{C}\mathbb{P}^m\). We describe the smooth nature and geometric structure of each strata component, generalizing the geometries of the embedded orbits in the model. This includes a quasi-Kähler metric on the open strata components that becomes singular at the strata boundary. The closed strata inherit almost CR-structures and can be viewed as a c-projective infinity for the given quasi-Kähler metric.

On the Shifted Littlewood–Richardson Coefficients and the Littlewood–Richardson Coefficients

We give a new interpretation of the shifted Littlewood–Richardson coefficients \(f_{\lambda \mu }^\nu \) (\(\lambda ,\mu ,\nu \) are strict partitions). The coefficients \(g_{\lambda \mu }\) which appear in the decomposition of Schur Q-function \(Q_\lambda \) into the sum of Schur functions \(Q_\lambda = 2^{l(\lambda )}\sum \nolimits _{\mu }g_{\lambda \mu }s_\mu \) can be considered as a special case of \(f_{\lambda \mu }^\nu \) (here \(\lambda \) is a strict partition of length \(l(\lambda )\)). We also give another description for \(g_{\lambda \mu }\) as the cardinal of a subset of a set that counts Littlewood–Richardson coefficients \(c_{\mu ^t\mu }^{{\tilde{\lambda }}}\). This new point of view allows us to establish connections between \(g_{\lambda \mu }\) and \(c_{\mu ^t \mu }^{{\tilde{\lambda }}}\). More precisely, we prove that \(g_{\lambda \mu }=g_{\lambda \mu ^t}\), and \(g_{\lambda \mu } \le c_{\mu ^t\mu }^{{\tilde{\lambda }}}\). We conjecture that \(g_{\lambda \mu }^2 \le c^{{\tilde{\lambda }}}_{\mu ^t\mu }\) and formulate some conjectures on our combinatorial models which would imply this inequality if it is valid.

Quantum W1+∞ subalgebras of BCD type and symmetric polynomials

The infinite affine Lie algebras of type ABCD, also called gl̂(∞), ô(∞), and sp̂(∞), are equivalent to subalgebras of the quantum W1+∞ algebras. They have well-known representations on the Fock space of a Dirac fermion (Â∞), a Majorana fermion (B̂∞ and D̂∞), or a symplectic boson (Ĉ∞). Explicit formulas for the action of the quantum W1+∞ subalgebras on the Fock states are proposed for each representation. These formulas are the equivalent of the vertical presentation of the quantum toroidal gl(1) algebra Fock representation. They provide an alternative to the fermionic and bosonic expressions of the horizontal presentation. Furthermore, these algebras are known to have a deep connection with symmetric polynomials. The action of the quantum W1+∞ generators leads to the derivation of Pieri-like rules and q-difference equations for these polynomials. In the specific case of B̂∞, a q-difference equation is obtained for Q-Schur polynomials indexed by strict partitions.

Beck-type companion identities for Franklin's identity via a modular refinement

The original Beck conjecture, now a theorem due to Andrews, states that the difference in the number of parts in all partitions into odd parts and the number of parts in all strict partitions is equal to the number of partitions whose set of even parts has one element, and also to the number of partitions with exactly one part repeated. This is a companion identity to Euler's identity. The theorem has been generalized by Yang to a companion identity to Glaisher's identity. Franklin generalized Glaisher's identity, and in this article, we provide a Beck-type companion identity for Franklin's identity and prove it via a modular refinement. We provide both analytical and combinatorial proofs. Andrews' and Yang's respective theorems fit naturally into this very general frame. We also give a generalization to Franklin's identity of the second Beck-type companion identity proved by Andrews and Yang in their respective work.

Bilinear expansions of lattices of KP τ -functions in BKP τ -functions: A fermionic approach

We derive a bilinear expansion expressing elements of a lattice of Kadomtsev-Petviashvili (KP) τ-functions, labeled by partitions, as a sum over products of pairs of elements of an associated lattice of BKP τ-functions, labeled by strict partitions. This generalizes earlier results relating determinants and Pfaffians of minors of skew symmetric matrices, with applications to Schur functions and Schur Q-functions. It is deduced using the representations of KP and BKP τ-functions as vacuum expectation values (VEVs) of products of fermionic operators of charged and neutral type, respectively. The lattice is generated by the insertion of products of pairs of charged creation and annihilation operators. The result follows from expanding the product as a sum of monomials in the neutral fermionic generators and applying a factorization theorem for VEVs of products of operators in the mutually commuting subalgebras. Applications include the case of inhomogeneous polynomial τ-functions of KP and BKP type.

New hook-content formulas for strict partitions

We introduce the difference operator for functions defined on strict partitions and prove a polynomiality property for a summation involving the bar length (hook length) and content statistics. As an application, several new hook-content formulas for strict partitions are derived.

Cut-and-join structure and integrability for spin Hurwitz numbers

Spin Hurwitz numbers are related to characters of the Sergeev group, which are the expansion coefficients of the Q Schur functions, depending on odd times and on a subset of all Young diagrams. These characters involve two dual subsets: the odd partitions (OP) and the strict partitions (SP). The Q Schur functions Q_R with Rin hbox {SP} are common eigenfunctions of cut-and-join operators W_Delta with Delta in hbox {OP}. The eigenvalues of these operators are the generalized Sergeev characters, their algebra is isomorphic to the algebra of Q Schur functions. Similarly to the case of the ordinary Hurwitz numbers, the generating function of spin Hurwitz numbers is a tau -function of an integrable hierarchy, that is, of the BKP type. At last, we discuss relations of the Sergeev characters with matrix models.

Random strict partitions and random shifted tableaux

We study asymptotics of random shifted Young diagrams which correspond to a given sequence of reducible projective representations of the symmetric groups. We show limit results (Law of Large Numbers and Central Limit Theorem) for their shapes, provided that the representation character ratios and their cumulants converge to zero at some prescribed speed. Our class of examples includes uniformly random shifted standard tableaux with prescribed shape as well as shifted tableaux generated by some natural combinatorial algorithms (such as shifted Robinson–Schensted–Knuth correspondence) applied to a random input.

On certain unimodal sequences and strict partitions

Building on a bijection of Vandervelde, we enumerate certain unimodal sequences whose alternating sum equals zero. This enables us to refine the enumeration of strict partitions with respect to the number of parts and the BG-rank.

Polynomiality of Plancherel averages of hook-content summations for strict, doubled distinct and self-conjugate partitions

The polynomiality of shifted Plancherel averages for summations of contents of strict partitions were established by the authors and Matsumoto independently in 2015, which is the key to determining the limit shape of random shifted Young diagram, as explained by Matsumoto. In this paper, we prove the polynomiality of shifted Plancherel averages for summations of hook lengths of strict partitions, which is an analog of the authors and Matsumoto's results on contents of strict partitions.In 2009, the first author proved the Nekrasov-Okounkov formula on hook lengths for integer partitions by using an identity of Macdonald in the framework of type A˜ affine root systems, and conjectured that the Plancherel averages of some summations over the set of all partitions of size n are always polynomials in n. This conjecture was generalized and proved by Stanley. Recently, Pétréolle derived two Nekrasov-Okounkov type formulas for C˜ and C˜ˇ which involve doubled distinct and self-conjugate partitions. Inspired by all those previous works, we establish the polynomiality of t-Plancherel averages of some hook-content summations for doubled distinct and self-conjugate partitions.

The Periodic Schur Process and Free Fermions at Finite Temperature

We revisit the periodic Schur process introduced by Borodin in 2007. Our contribution is threefold. First, we provide a new simpler derivation of its correlation functions via the free fermion formalism. In particular, we shall see that the process becomes determinantal by passing to the grand canonical ensemble, which gives a physical explanation to Borodin's "shift-mixing" trick. Second, we consider the edge scaling limit in the simplest nontrivial case, corresponding to a deformation of the poissonized Plancherel measure on partitions. We show that the edge behavior is described, in a certain crossover regime different from that for the bulk, by the universal finite-temperature Airy kernel, which was previously encountered by Johansson and Le Doussal et al. in other models, and whose extreme value statistics interpolates between the Tracy-Widom GUE and the Gumbel distributions. We also define and prove convergence for a stationary extension of our model. Finally, we compute the correlation functions for a variant of the periodic Schur process involving strict partitions, Schur's P and Q functions, and neutral fermions.

BKP hierarchy and Pfaffian point process

Inspired by Okounkov's work (2001) [20] which relates KP hierarchy to determinant point process, we establish a relationship between BKP hierarchy and Pfaffian point process. We prove that the correlation function of the shifted Schur measures on strict partitions can be expressed as a Pfaffian of skew symmetric matrix kernel, whose elements are certain vacuum expectations of neutral fermions. We further show that the matrix integrals solution of BKP hierarchy can also induce a certain Pfaffian point process.

The butterfly sequence: the second difference sequence of the numbers of integer partitions with distinct parts

We interpret the second difference of the sequence of the number of strict partitions, for n≥5, as the sequence of the strict partitions of n with at least three parts, the three largest parts consecutive, and the smallest part at least two. The name butterfly describes both a sequence's interpretation and a bijection between subsets of strict partitions. Using cyclotomic polynomials, we compute generating function identities of the butterfly sequence and related sequences both as infinite products and as series filtered by the number of parts. We offer a merging and splitting construction of the butterfly sequence as a sequence of partitions with odd parts larger or equal to 3, and we interpret the butterfly sequence as a sequence of generalized pentagonal, pentagonal with domino, and non-pentagonal butterfly partitions. Euler's Pentagonal Number Theorem and a similar specialization of the Jacobi Triple Product lead to recursive algorithms to compute the butterfly sequence and related sequences.

A Spin Analogue of Kerov Polynomials

Kerov polynomials describe normalized irreducible characters of the symmetric groups in terms of the free cumulants associated with Young diagrams. We suggest well-suited counterparts of the Kerov polynomials in spin (or projective) representation settings. We show that spin analogues of irreducible characters are polynomials in even free cumulants associated with double diagrams of strict partitions. Moreover, we present a conjecture for the positivity of their coefficients.

New hook-content formulas for strict partitions

We introduce the difference operator for functions defined on strict partitions and prove a polynomiality property for a summation involving the hook length and content statistics. As an application, several new hook-content formulas for strict partitions are derived.

Polynomial approach to explicit formulae for generalized binomial coefficients

We extend the polynomial approach to the hook length formula proposed in Karolyi et al. Adv Math 277:252–282 (2015) to several other problems of the same type, including the number of paths formula in the Young graph of strict partitions.

The Andrews–Olsson identity and Bessenrodt insertion algorithm on Young walls

We extend the Andrews–Olsson identity to two-colored partitions. Regarding the sets of proper Young walls of quantum affine algebras gn=A2n(2), A2n−1(2), Bn(1), Dn(1) and Dn+1(2) as the sets of two-colored partitions, the extended Andrews–Olsson identity implies that the generating functions of the sets of reduced Young walls have very simple formulae:∏i=1∞(1+ti)κiwhere κi=0,1 or 2, andκi varies periodically. Moreover, we generalize Bessenrodt’s algorithms to prove the extended Andrews–Olsson identity in an alternative way. From these algorithms, we can give crystal structures on certain subsets of pair of strict partitions which are isomorphic to the crystal bases B(Λ) of the level 1 highest weight modules V(Λ) over Uq(gn).