The projective degrees of strict partitions of n were computed for all n < 101 and the partitions with maximal projective degree were found for each n. It was observed that maximizing partitions for successive values of n "lie close to each other" in a certain sense. Conjecturing that this holds for larger values of n, the partitions of maximal degree were computed for all n < 221. The results are consistent with a recent conjecture on the limiting shape of the strict partition of maximal projective degree.
We determine the partition dimension of the incidence graph $G(Π_q)$ of the projective plane $Π_q$ up to a constant factor $2$ as $(2+o(1))\log_2{q}\leq \mathrm{pd}(G(Π_q))\leq (4+o(1))\log_2{q}.$
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.
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.
The z-measures on partitions originated from the problem of harmonic analysis of linear representations of the infinite symmetric group in the works of Kerov, Olshanski and Vershik (1993, 2004). A similar family corresponding to projective representations was introduced by Borodin (1997). The latter measures live on strict partitions (i.e., partitions with distinct parts), and the z-measures are supported by all partitions. In this note we describe some combinatorial relations between these two families of measures using the well-known doubling of shifted Young diagrams.
Let $L=\mathbb F_{q^n}$ be a finite field and let $F=\mathbb F_q$ be a subfield of $L$. Consider $L$ as a vector space over $F$ and the associated projective space that is isomorphic to ${\mathrm{PG}}(n-1,q)$. The properties of the projective mapping induced by $x\mapsto x^{-1}$ have been studied in \cite{Cs13,Fa02,Ha83,He85,Bu95}, where it is proved that the image of any line is a normal rational curve in some subspace. In this note a more detailed geometric description is achieved. Consequences are found related to mixed partitions of the projective spaces; in particular, it is proved that f…
Bessenrodt and Ono initially found the strict log-subadditivity of partition function $p(n)$, that is, $p(a+b)< p(a)p(b)$ for $a,b>1$ and $a+b>9$. Many other important statistics of partitions are proved to enjoy similar properties. Lovejoy introduced the overpartition rank as an analog of Dyson's rank for partitions from the $q$-series perspective. Let $\overline{N}(a,c,n)$ denote the number of overpartitions with rank congruent to $a$ modulo $c$. Ciolan computed the asymptotic formula of $\overline{N}(a,c,n)$ and showed that $\overline{N}(a, c, n) > \overline{N}(b, c, n)$ for $c\geq7$ and $n…
Regarding Euler's odd-strict theorem, which is the most basic partition identity, A refinement was done by Sylvester, and it was generalized by Bessenrodt to the r-regular and r-class regular cases. In this paper, we focus on the periodicity of the exponent seen on the r-regular side in Bessenrodt's generalization, and further generalize from that point of view. At the end of this paper, we also introduce a partition identity when extended without regard to the periodicity of the exponent.
We construct weight-preserving bijections between column strict shifted plane partitions with one row and alternating sign trapezoids with exactly one column in the left half that sums to $1$. Amongst other things, they relate the number of $-1$s in the alternating sign trapezoids to certain elements in the column strict shifted plane partitions that generalise the notion of special parts in descending plane partitions. The advantage of these bijections is that they include configurations with $-1$s, which is a feature that many of the bijections in the realm of alternating sign arrays lack.
AG codes correspond geometrically to points in the Grassmannian of k-planes in an n-dimensional projective space PG(n, F_q) defined over a finite field F_q of q elements. We prove that invariant subgrassmannians by the action of a triangle group hold a t-design of determined parameters.
A vector space partition $\mathcal{P}$ of the projective space $\operatorname{PG}(v-1,q)$ is a set of subspaces in $\operatorname{PG}(v-1,q)$ which partitions the set of points. We say that a vector space partition $\mathcal{P}$ has type $(v-1)^{m_{v-1}} \dots 2^{m_2}1^{m_1}$ if precisely $m_i$ of its elements have dimension $i$, where $1\le i\le v-1$. Here we determine all possible types of vector space partitions in $\operatorname{PG}(7,2)$.
We study a family of continuous time Markov jump processes on strict partitions (partitions with distinct parts) preserving the distributions introduced by Borodin (1997) in connection with projective representations of the infinite symmetric group. The one-dimensional distributions of the processes (i.e., the Borodin's measures) have determinantal structure. We express the dynamical correlation functions of the processes in terms of certain Pfaffians and give explicit formulas for both the static and dynamical correlation kernels using the Gauss hypergeometric function. Moreover, we are able …
This article extends recent results on log-Coulomb gases in a $p$-field $K$ (i.e., a nonarchimedean local field) to those in its projective line $\mathbb{P}^1(K)$, where the latter is endowed with the $PGL_2$-invariant Borel probability measure and spherical metric. Our first main result is an explicit combinatorial formula for the canonical partition function of log-Coulomb gases in $\mathbb{P}^1(K)$ with arbitrary charge values. Our second main result is called the "$(q+1)$th Power Law", which relates the grand canonical partition functions for one-component gases in $\mathbb{P}^1(K)$ (where…
We study a curious class of partitions, the parts of which obey an exceedingly strict congruence condition we refer to as "sequential congruence": the $m$th part is congruent to the $(m+1)$th part modulo $m$, with the smallest part congruent to zero modulo the length of the partition. It turns out these obscure-seeming objects are embedded in a natural way in partition theory. We show that sequentially congruent partitions with largest part $n$ are in bijection with the partitions of $n$. Moreover, we show sequentially congruent partitions induce a bijection between partitions of $n$ and parti…
We study the restrictions, the strict fixed points, and the strict quotients of the partition complex $|Π_n|$, which is the $Σ_n$-space attached to the poset of proper nontrivial partitions of the set $\{1,\ldots,n\}$. We express the space of fixed points $|Π_n|^G$ in terms of subgroup posets for general $G\subset Σ_n$ and prove a formula for the restriction of $|Π_n|$ to Young subgroups $Σ_{n_1}\times \dots\times Σ_{n_k}$. Both results follow by applying a general method, proven with discrete Morse theory, for producing equivariant branching rules on lattices with group actions. We uncover su…
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 t…
In this paper, we obtain infinitely many non-trivial identities and inequalities between full rank differences for 2-marked Durfee symbols, a generalization of partitions introduced by Andrews. A certain strict inequality, which almost always holds, shows that identities for Dyson's rank, similar to those proven by Atkin and Swinnerton-Dyer, are quite rare. By showing an analogous strict inequality, we show that such non-trivial identities are also rare for the full rank, but on the other hand we obtain an infinite family of non-trivial identities, contrasting the partition theoretic case.
