The Haglund–Haiman–Loehr theorem provides the following combinatorial formula for the modified Macdonald polynomials: H̃_μ(X;q,t)=∑ _σ : μ→ℙx^σt^(σ )q^(σ ). Inspired by Martin’s multiline-queue formula for the stationary distribution of multitype asymmetric simple exclusion processes, Corteel, Haglund, Mandelshtam, Mason and Williams recently introduced the queue inversion statistic and conjectured that the tableaux formula for H̃_μ(X;q,t) is invariant if the inversion statistic is replaced by . This was subsequently resolved by Ayyer, Mandelshtam and Martin, who proposed a stronger conjecture on the equivalence of the two refined formulas for H̃_μ(X;q,t) . Our main result confirms this Ayyer–Mandelshtam–Martin conjecture. We establish an equidistribution between the pairs (,) and (,) of μ -Mahonian statistics on any row-equivalency class [τ ] , where τ is a filling of the Young diagram of μ . As a byproduct of our approach, we show that if τ is a rectangular filling, the triples (,,) and (,,) have the same distribution over [τ ] .

The question of classifying when two skew Schur functions are equal is a substantial open problem, which remains unsolved for over a century. In 2022, Aliniaeifard, Li and van Willigenburg introduced skew Schur functions in noncommuting variables, $s_{(\delta,D)}$, where $D$ is a connected skew diagram with $n$ boxes and $\delta$ is a permutation in the symmetric group $S_n$. In this paper, we combine these two and classify when two skew Schur functions in noncommuting variables are equal: $s_{(\delta,D)} = s_{(\tau,T)}$ such that $D\ne T$ if and only if $D$ is a nonsymmetric ribbon, $T$ is the antipodal rotation of $D$ and $\overline{\tau^{-1}\delta}$ is an explicit bijection between two set partitions determined by $D$.

Two skew diagrams are defined to be equivalent if their corresponding skew Schur functions are equal. The equivalence classes of ribbons (edgewise connected skew diagrams without a 2× 2 block of boxes) have been classified by Billera, Thomas and van Willigenburg in 2006. In this paper, we provide a complete characterization of the equivalence classes of connected skew diagrams with exactly one inclusion-maximal 2× m or m× 2 block of boxes for every m≥ 2. In particular, the possible sizes of such equivalence classes are one, two, or four, demonstrating that a single 2× m or m× 2 block dramatically reduces the sizes of equivalence classes. Our result confirms special cases of the elusive conjecture on equivalent connected skew diagrams proposed by McNamara and van Willigenburg in 2009.

We establish the asymptotic normality of the dimension of large-size random Fishburn matrices by a complex-analytic approach. The corresponding dual problem of size distribution under large dimension is also addressed and follows a quadratic type normal limit law. These results represent the first of their kind and solve two open questions raised in the combinatorial literature. They are presented in a general framework where the entries of the Fishburn matrices are not limited to {0, 1} or nonnegative integers N-0. The analytic saddle-point approach we apply, based on a powerful transformation for q-series due to Andrews and Jelinek, is also useful in solving a conjecture of Stoimenow in Vassiliev invariants.

We present (bi-)symmetric generating functions for the joint distributions of Euler–Stirling statistics on permutations, including the number of descents (des), inverse descents (ides), the number of left-to-right maxima (lmax), the number of right-to-left maxima (rmax) and the number of left-to-right minima (lmin). We also show how they recover the classical symmetric generating function for permutations due to Carlitz, Roselle and Scoville (1966). Our proofs exploit three different recursive constructions of inversion sequences, bijections on the multiple equidistributions of Euler–Stirling statistics over permutations and transformation formulas of basic hypergeometric series. We further establish a new quadruple equidistribution of Euler–Stirling statistics over inversion sequences, as progress towards a conjecture proposed by Schlosser and the author (2020).

It is well known since the seminal work by Bousquet-Melou, Claesson, Dukes and Kitaev (2010) that certain refinements of the ascent sequences with respect to several natural statistics are in bijection with corresponding refinements of (2 + 2)-free posets and permutations that avoid a bi-vincular pattern. Different multiply-refined enumerations of ascent sequences and other bijectively equivalent structures have subsequently been extensively studied by various authors. In this paper, our main contributions are center dot a bijective proof of a bi-symmetric septuple equidistribution of Euler-Stirling statistics on ascent sequences, involving the number of ascents (asc), the number of repeated entries (rep), the number of zeros (zero), the number of maximal entries (max), the number of right-to-left minima (rmin) and two auxiliary statistics; center dot a new transformation formula for non-terminating basic hypergeometric 4 phi 3 series expanded as an analytic function in base q around q = 1, which is utilized to prove two (bi)-symmetric quadruple equidistributions on ascent sequences. A by-product of our findings includes the affirmation of a conjecture about the bi-symmetric equidistribution between the quadruples of Euler-Stirling statistics (asc, rep, zero, max) and (rep, asc, max, zero) on ascent sequences, that was motivated by a double Eulerian equidistribution due to Foata (1977) and recently proposed by Fu, Lin, Yan, Zhou and the first author (2018).

A direct saddle-point analysis (without relying on any modular forms or functional equations) is developed to establish the asymptotics of Fishburn matrices and a large number of other variants with a similar sum-of-finite-product form for their (formal) generating functions. In addition to solving some conjectures, the application of our saddle-point approach to the distributional aspects of statistics on Fishburn matrices is also examined with many new limit theorems characterized, representing the first of their kind for such structures.

We study random unlabelled $k$-dimensional trees by combining the colouring approach by Gainer-Dewar and Gessel (2014) with the cycle pointing method by Bodirsky, Fusy, Kang and Vigerske (2011). Our main applications are Gromov-Hausdorff-Prokhorov and Benjamini-Schramm limits, that describe their asymptotic geometric shape on a global and local scale as the number of hedra tends to infinity.

As shown by Bousquet-Melou-Claesson-Dukes-Kitaev (2010), ascent sequences can be used to encode (2 + 2)-free posets. It is known that ascent sequences are enumerated by the Fishburn numbers, which appear as the coefficients of the formal power series Sigma(infinity)(m=1)Pi(m)(i=1) (1 - (1 - t)i). In this paper, we present a novel way to recursively decompose ascent sequences, which leads to: a calculation of the Euler-Stirling distribution on ascent sequences, including the numbers of ascents (asc), repeated entries (rep), zeros (zero) and maximal entries (max). In particular, this confirms and extends Dukes and Parviainen's conjecture on the equidistribution of zero and max. a far-reaching generalization of the generating function formula for (asc, zero) due to Jelinek. This is accomplished via a bijective proof of the quadruple equidistribution of (asc, rep, zero, max) and (rep, asc, rmin, zero), where rmin denotes the right-to-left minima statistic of ascent sequences. an extension of a conjecture posed by Levande, which asserts that the pair (asc, zero) on ascent sequences has the same distribution as the pair (rep, max) on (2 - 1)-avoiding inversion sequences. This is achieved via a decomposition of (2 - 1)-avoiding inversion sequences parallel to that of ascent sequences. This work is motivated by a double Eulerian equidistribution of Foata (1977) and a tempting bi-symmetry conjecture, which asserts that the quadruples (asc, rep, zero, max) and (rep, asc, max, zero) are equidistributed on ascent sequences. (C) 2019 Elsevier Inc. All rights reserved.

Inversion sequences are in natural bijection with permutations and have surprising connections with lecture hall polytopes and partitions. Recently, Martinez and Savage carried out the systematic study of inversion sequences avoiding triples of relations. They established many connections with known integer sequences and highlighted several interesting conjectures, some of which have already been solved. In this paper, we address the remaining enumeration conjectures posed by them, leaving only those related to the OEIS sequence A098746 open.

For any set Ω of non‐negative integers such that , we consider a random Ω‐k‐tree Gn,k that is uniformly selected from all connected k‐trees of (n + k) vertices such that the number of (k + 1)‐cliques that contain any fixed k‐clique belongs to Ω. We prove that Gn,k, scaled by where Hk is the kth harmonic number and σΩ > 0, converges to the continuum random tree . Furthermore, we prove local convergence of the random Ω‐k‐tree to an infinite but locally finite random Ω‐k‐tree G∞,k.

In this paper we describe the thickened strips and the outside nested decompositions of any skew shape $\lambda/\mu$. For any such decomposition $\Phi=(\Theta_1,\Theta_2,\ldots,\Theta_g)$ of the skew shape $\lambda/\mu$ where $\Theta_i$ is a thickened strip for every $i$, if $r$ is the number of boxes that are contained in any two distinct thickened strips of $\Phi$, we establish a determinantal formula of the function $s_{\lambda/\mu}(X)p_{1^r}(X)$ with the Schur functions of thickened strips as entries, where $s_{\lambda/\mu}(X)$ is the Schur function of the skew shape $\lambda/\mu$ and $p_{1^r}(X)$ is the power sum symmetric function index by the partition $(1^r)$. This generalizes Hamel and Goulden's theorem on the outside decompositions of the skew shape $\lambda/\mu$. As an application of our theorem, we derive the number of $m$-strip tableaux which was first counted by Baryshnikov and Romik via extending the transfer operator approach due to Elkies.

Panagiotou and Stufler recently proved an important fact on their way to establish the scaling limits of random Polya trees: a uniform random Polya tree of size n consists of a conditioned critical Galton-Watson tree C-n and many small forests, where with probability tending to one, as n tends to infinity, any forest F-n(v), that is attached to a node v in C-n is maximally of size vertical bar F-n(v)vertical bar = O(log n). Their proof used the framework of a Boltzmann sampler and deviation inequalities. In this paper, first, we employ a unified framework in analytic combinatorics to prove this fact with additional improvements for vertical bar F-n(v)vertical bar, namely vertical bar F-n(v)vertical bar = Theta(log n). Second, we give a combinatorial interpretation of the rational weights of these forests and the defining substitution process in terms of automorphisms associated to a given Polya tree. Third, we derive the limit probability that for a random node v the attached forest F-n(v) is of a given size. Moreover, structural properties of those forests like the number of their components are studied. Finally, we extend all results to other Polya structures. (C) 2018 Elsevier B.V. All rights reserved.

Previous chapter Next chapter Full AccessProceedings 2016 Proceedings of the Meeting on Analytic Algorithmics and Combinatorics (ANALCO)Scaling limit of random k-treesMichael Drmota and Emma Yu JinMichael Drmota and Emma Yu Jinpp.56 - 65Chapter DOI:https://doi.org/10.1137/1.9781611974324.7PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstract We consider a random k-tree Gn,k that is uniformly selected from the class of labelled k-trees with n + k vertices. Since 1-trees are just trees, it is well-known that Gn,1 (after scaling the distances by converges to the Continuum Random Tree Our main result is that for k ≠ 1, the random k-tree Gn,k, scaled by where Hk–1 is the (k – 1)-th Harmonic number, converges to the Continuum Random Tree too. In particular this shows that the diameter as well as the typical distance of two vertices in a random k-tree Gn,k are of order Previous chapter Next chapter RelatedDetails Published:2016eISBN:978-1-61197-432-4 https://doi.org/10.1137/1.9781611974324Book Series Name:ProceedingsBook Code:PRAN16Book Pages:1-148

Take Q = (Q(1), Q(2), ...) to be an exponential structure and M(n) to be the number of minimal elements of Q(n) where M(0) = 1. Then a sequence of numbers {r(n)(Q(n))}(n >= 1) is defined by the equationSigma(n >= 1)r(n)(Q(n))z(n)/n!M(n) = -log (Sigma(n >= 0) (-1)(n) z(n)/n!M(n))Let (Q) over bar (n) denote the poset Q(n) with a (0) over cap adjoined and let (1) over cap denote the unique maximal element in the poset Q(n). Furthermore, let mu(Qn) be the Mobius function on the poset (Q) over bar (n). Stanley proved that r(n)(Q(n)) = (-1)(n)mu(Qn)((0) over cap, (1) over cap) This implies that the numbers r(n)(Q(n)) are integers. In this paper, we study the cases Q(n) = Pi((r))(n) and Q(n) = Q(n)((r)) where Pi((r))(n) and Q(n)((r)) are posets, respectively, of set partitions of [rn] whose block sizes are divisible by r and of r -partitions of [n]. In both cases we prove that r(n)(Pi((r))(n)) and r(n)(Q(n)((r))) enumerate the pyramids by applying the Cartier-Foata monoid identity and further prove that r(n)(Pi((r))(n)) is the generalized Euler number Ern-1 and that r(n)(Q(n)((2))) is the number of complete non-ambiguous trees of size 2n - 1 by bijections. This gives a new proof of Welker's theorem that r(n)(Pi((r))(n)) = Ern-1 and implies the construction of r-dimensional complete non-ambiguous trees. As a bonus of applying the theory of heaps, we establish a bijection between the set of complete non-ambiguous forests and the set of pairs of permutations with no common rise. This answers an open question raised by Aval et al. (C) 2015 Elsevier Ltd. All rights reserved.

In this paper we solve the asymptotic counting problemfor unlabeledk-trees. By applying a proper singularity analysis of generating functions we show that the num bersUn of unlabeledk-trees of sizen are asymptotically given byUn ∼ ckn(ρ1k) , whereck > 0 andρ1k > 0 denotes the radius of convergence of the generating functionU(z) = ∑ n≥0 Unz . Furthermore we prove that the number of leavesand more generally the number of nodesof given degree satisfy a central limit theorem with mean val ue and variance that are asymptotically linear in the number of hedra where a hedron is a (k + 1)-clique in ak-tree.

Various tools used to predict the secondary structure for a given RNA sequence are based on dynamic programming used to compute a conformation of minimum free energy. For structures without pseudoknots, a worst-case runtime proportional to n3, with n being the length of the sequence, results since a table of dimension n2 has to be filled in while a single entry gives rise to a linear computational effort. However, it was recently observed that reformulating the corresponding dynamic programming recursion together with the bookkeeping of potential folding alternatives (a technique called sparsification) may reduce the runtime to n2 on average, assuming that nucleotides of distance d form a hydrogen bond (i..e., are paired) with probability b/d(c) for some constants b > 0, c > 1. The latter is called the polymer-zeta model and plays a crucial role in speeding up the above mentioned algorithm. In this paper we discuss the application of the polymer-zeta property for the analysis of sparsification, showing that it must be applied conditionally on first and last positions to pair. Afterwards, we will investigate the combinatorics of RNA secondary structures assuming that the corresponding conditional probabilities behave according to a polymer-zeta probability model. We show that even if some of the structural parameters exhibit an almost realistic behavior on average, the expected shape of a folding in that model must be assumed to highly differ from those observed in nature. More precisely, we prove our polymer-zeta model to be appropriate for mRNA molecules but to fail in connection with almost every other family of RNA. Those findings explain the huge speedup of the dynamic programming algorithm observed empirically by Wexler et al. when applying sparsification in connection with mRNA data.

Baryshnikov and Romik derived the combinatorial identities for the numbers of the $m$-strip tableaux. This generalized the classical André's theorem for the number of up-down permutations. They asked for a bijective proof for the enumeration of $3$-strip tableaux. In this paper we will provide such a bijective proof. First we count the $3$-strip tableaux by decomposition. Secondly we will apply this "decomposition" idea on the up-down permutations and down-up permutations to enumerate the $3$-strip tableaux bijectively.

In this paper we prove a $q$-analogue of Koshy's formula in terms of the Narayana polynomial due to Lassalle and a $q$-analogue of Koshy's formula in terms of $q$-hypergeometric series due to Andrews by applying the inclusion-exclusion principle on Dyck paths and on partitions. We generalize these two $q$-analogues of Koshy's formula for $q$-Catalan numbers to that for $q$-Ballot numbers. This work also answers an open question by Lassalle and two questions raised by Andrews in 2010. We conjecture that if $n$ is odd, then for $m\ge n\ge 1$, the polynomial $(1+q^n){m\brack n-1}_q$ is unimodal. If $n$ is even, for any even $j\ne 0$ and $m\ge n\ge 1$, the polynomial $(1+q^n)[j]_q{m\brack n-1}_q$ is unimodal. This implies the answer to the second problem posed by Andrews.

The Shapley value and the fair proportion index of phylogenetic trees have been introduced recently for the purpose of making conservation decisions in genetics. Moreover, also very recently, Hartmann (J Math Biol 67:1163–1170, 2013) has presented data which shows that there is a strong correlation between a slightly modified version of the Shapley value (which we call the modified Shapley value) and the fair proportion index. He gave an explanation of this correlation by showing that the contribution of both indices to an edge of the tree becomes identical as the number of taxa tends to infinity. In this note, we show that the Shapley value and the fair proportion index are in fact the same. Moreover, we also consider the modified Shapley value and show that its covariance with the fair proportion index in random phylogenetic trees under the Yule-Harding model and uniform model is indeed close to one.