Ballot Sequences Research Articles

A determination of Catalan numbers in 18th century Italy by Giovanni Rizzetti (1675–1751)

We discuss the probabilistic question faced by the Italian scholar Giovanni Rizzetti (1675–1751) and suggest this is a variant of what we refer to today as the “ballot sequences”, variant to which Catalan numbers offer a solution.Rizzetti's search for a solution led him to produce an iterative rule which involved the consideration of Catalan numbers. Although coming up with these were not his final goal, the rule he elaborated allowed him to carry on calculating them as long as he wanted. As such, it is possible this was the first time such numbers were determined.

Quasisymmetric harmonics of the exterior algebra

AbstractWe study the ring of quasisymmetric polynomials in n anticommuting (fermionic) variables. Let $R_n$ denote the ring of polynomials in n anticommuting variables. The main results of this paper show the following interesting facts about quasisymmetric polynomials in anticommuting variables: (1)The quasisymmetric polynomials in $R_n$ form a commutative subalgebra of $R_n$ .(2)There is a basis of the quotient of $R_n$ by the ideal $I_n$ generated by the quasisymmetric polynomials in $R_n$ that is indexed by ballot sequences. The Hilbert series of the quotient is given by $$ \begin{align*}\text{Hilb}_{R_n/I_n}(q) = \sum_{k=0}^{\lfloor{n/2}\rfloor} f^{(n-k,k)} q^k\,,\end{align*} $$ where $f^{(n-k,k)}$ is the number of standard tableaux of shape $(n-k,k)$ .(3)There is a basis of the ideal generated by quasisymmetric polynomials that is indexed by sequences that break the ballot condition.

Generating 2-Gray codes for ballot sequences in constant amortized time

We present a simple algorithm that generates a cyclic 2-Gray code for ballot sequences. The algorithm generates each ballot sequence in constant amortized time using a linear amount of space. This is the first known cyclic 2-Gray code for ballot sequences that achieves this time bound. In addition, the algorithm can be easily modified to output ballot sequences in binary reflected Gray code order in constant amortized time per string using a linear amount of space.

On the critical exponents of generalized ballot sequences in three dimensions and large tandem walks

We answer some questions on the asymptotics of ballot walks raised in [S. B. Ekhad and D. Zeilberger, April 2021] and prove that these models are not D-finite. This short note demonstrates how the powerful tools developed in the last decades on lattice paths in convex cones help us to answer some challenging problems that were out of reach for a long time. On the way we generalize tandem walks to the family of large tandem walks whose steps are of arbitrary length and map them bijectively to a generalization of ballot walks in three dimensions.

Pattern Avoiding Permutations with a Unique Longest Increasing Subsequence

We investigate permutations and involutions that avoid a pattern of length three and have a unique longest increasing subsequence (ULIS). We prove an explicit formula for 231-avoiders, we show that the growth rate for 321-avoiding permutations with a ULIS is 4, and prove that their generating function is not rational. We relate the case of 132-avoiders to the existing literature, raising some interesting questions. For involutions, we construct a bijection between 132-avoiding involutions with a ULIS and bidirectional ballot sequences.

On the exhaustive generation of generalized ballot sequences in lexicographic and Gray code order

Abstract A generalized (resp. p-ary) ballot sequence is a sequence over the set of non-negative integers (resp. integers less than p) where in any of its prefixes each positive integer i occurs at most as often as any integer less than i. We show that the Reected Gray Code order induces a cyclic 3-adjacent Gray code on both, the set of fixed length generalized ballot sequences and p-ary ballot sequences when p is even, that is, ordered list where consecutive sequences (regarding the list cyclically) differ in at most 3 adjacent positions. Non-trivial efficient generating algorithms for these ballot sequences, in lexicographic order and for the obtained Gray codes, are also presented.

Counting arithmetical structures on paths and cycles

Let G be a finite, connected graph. An arithmetical structure on G is a pair of positive integer vectors d,r such that (diag(d)−A)r=0, where A is the adjacency matrix of G. We investigate the combinatorics of arithmetical structures on path and cycle graphs, as well as the associated critical groups (the torsion part of the cokernels of the matrices (diag(d)−A)). For paths, we prove that arithmetical structures are enumerated by the Catalan numbers, and we obtain refined enumeration results related to ballot sequences. For cycles, we prove that arithmetical structures are enumerated by the binomial coefficients 2n−1n−1, and we obtain refined enumeration results related to multisets. In addition, we determine the critical groups for all arithmetical structures on paths and cycles.

Analysis of Bidirectional Ballot Sequences and Random Walks Ending in Their Maximum

Consider non-negative lattice paths ending at their maximum height, which will be called admissible paths. We show that the probability for a lattice path to be admissible is related to the Chebyshev polynomials of the first or second kind, depending on whether the lattice path is defined with a reflective barrier or not. Parameters like the number of admissible paths with given length or the expected height are analyzed asymptotically. Additionally, we use a bijection between admissible random walks and special binary sequences to prove a recent conjecture by Zhao on ballot sequences.

Knights, spies, games and ballot sequences

This paper solves the Knights and Spies Problem: In a room there are n people, each labelled with a unique number between 1 and n . A person may either be a knight or a spy. Knights always tell the truth, while spies may lie or tell the truth as they see fit. Each person in the room knows the identity of everyone else. Apart from this, all that is known is that strictly more knights than spies are present. Asking only questions of the form: ‘Person i , what is the identity of person j ?’, what is the least number of questions that will guarantee to find the true identities of all n people? We present a questioning strategy that uses slightly less than 3 n / 2 questions, and prove that it is optimal by solving a related two-player game. The performance of this strategy is analysed using methods from the famous ballot-counting problem. We end by discussing two questions suggested by generalisations of the original problem.

Constructing MSTD sets using bidirectional ballot sequences

A more sums than differences (MSTD) set is a finite subset S of the integers such that | S + S | > | S − S | . We construct a new dense family of MSTD subsets of { 0 , 1 , 2 , … , n − 1 } . Our construction gives Θ ( 2 n / n ) MSTD sets, improving the previous best construction with Ω ( 2 n / n 4 ) MSTD sets by Miller, Orosz, and Scheinerman.

Enumerative properties of shifted-Dyck partitions

We study the enumerative properties of a new class of (skew) shifted partitions. This class arises in the computation of certain parabolic Kazhdan–Lusztig polynomials and is closely related to ballot sequences. As consequences of our results, we obtain new identities for the parabolic Kazhdan–Lusztig polynomials of Hermitian symmetric pairs and for the ordinary Kazhdan–Lusztig polynomials of certain Weyl groups.

Creation of Ballot Sequences in a Periodic Cellular Automaton

Motivated by an attempt to develop a method for solving initial value problems in a class of one dimensional periodic cellular automata (CA) associated with crystal bases and soliton equations, we consider a generalization of a simple proposition in elementary mathematics. The original proposition says that any sequence of letters 1 and 2, having no less 1's than 2's, can be changed into a ballot sequence via cyclic shifts only. We generalize it to treat sequences of cells of common capacity s > 1, each of them containing consecutive 2's (left) and 1's (right), and show that these sequences can be changed into a ballot sequence via two manipulations, cyclic and quasi-cyclic shifts. The latter is a new CA rule and we find that various kink-like structures are traveling along the system like particles under the time evolution of this rule.

Enumeration via ballot numbers

Several interesting combinatorial coefficients such as the Catalan numbers and the Bell numbers can be described either via a 3-term recurrence or as sums of (weighted) ballot numbers. This paper gives some general results connecting 3-term recurrences with ballot sequences with several applications to the enumeration of various combinatorial instances.

On the generation ofP-sequences

An efficient algorithm for generating the P-sequences characterising all shapes of binary trees with n nodes is presented. This is based on the connection between the P-sequences and the ballot sequences. A P-sequence results from a ballot sequence if the integers 0, 1, 2,…,n−1 in ballot sequence is replaced by integers n, n−1, n−2, …, 1 in their corresponding positions.

Generalized Dyck paths

It is well known (see [3, 6, 9, 10, 11]) that Dyck paths are in bijection with “Dyck words”, “ballot sequences”, “well formed sequences of parentheses”, “2-lines standard-tableaux”, “binary trees”, “ordered trees”; all these are counted by Catalan numbers. In the present text, we replace the north-east steps of a Dyck path by steps from an arbitrary finite multi-set l of vectors with integral coordinates in the plane. In order to study these generalized Dyck paths, called A-paths, we have to introduce many closely related families of paths. The corresponding (multi-variable) generating functions satisfy an intricate system of algebraic equations which leads to a polynomial equation satisfied by A . For example when l is {(1, 1)} (respectively {(1,2), (2,1); {(1,3), (3,1)}) this polynomial equation is of degree 2 (resp. 4, 8). More generally when l ={u 1, u 2,…, u m} where u j = ( r j , j) then A = A( u 1, u 2,…, u m ) satisfies a polynomial equation of degree 2 m with coefficients in Z [ u 1, u 2,…, u m ].

Ballot sequences and a determinant of good's

Ballot sequences and a determinant of good's

Generation of Binary Trees from Ballot Sequences

Generation of Binary Trees from Ballot Sequences