Distance Fibonacci Polynomials by Graph Methods

In this paper we introduce and study a new generalization of Fibonacci polynomials which generalize Fibonacci, Jacobsthal and Narayana numbers, simultaneously. We give a graph interpretation of these polynomials and we obtain a binomial formula for them. Moreover by modification of Pascal’s triangle, which has a symmetric structure, we obtain matrices generated by coefficients of generalized Fibonacci polynomials. As a consequence, the direct formula for generalized Fibonacci polynomials was given. In addition, we determine matrix generators for generalized Fibonacci polynomials, using the symmetric matrix of initial conditions.

A Detailed Study on Matrix Sequence of Generalized Narayana Numbers

In this paper, we introduce and investigate the generalized Narayana matrix sequence and we deal with, in detail, three special cases of this sequence which we call them Narayana, Narayana-Lucas and Narayana-Perrin matrix sequences. We present Binet’s formulas, generating functions, and the summation formulas for these sequences. We present the proofs to indicate how these sum formulas, in general, were discovered. Of course, all the listed sum formulas may be proved by induction, but that method of proof gives no clue about their discovery. Moreover, we give some identities and matrices related with these sequences. Furthermore, we show that there always exist interrelation between generalized Narayana, Narayana, Narayana-Lucas and Narayana-Perrin matrix sequences.

Several Determinantal Expressions of Generalized Tribonacci Polynomials and Sequences

In the paper, the authors present several explicit formulas for the $(p,q,r)$-Tribonacci polynomials and generalized Tribonacci sequences in terms of the Hessenberg determinants and, consequently, derive several explicit formulas for the Tribonacci numbers and polynomials, the Tribonacci--Lucas numbers, the Perrin numbers, the Padovan (Cordonnier) numbers, the Van der Laan numbers, the Narayana numbers, the third order Jacobsthal numbers, and the third order Jacobsthal--Lucas numbers in terms of special Hessenberg determinants.

On (k,p)-Fibonacci Numbers

In this paper, we introduce and study a new two-parameters generalization of the Fibonacci numbers, which generalizes Fibonacci numbers, Pell numbers, and Narayana numbers, simultaneously. We prove some identities which generalize well-known relations for Fibonacci numbers, Pell numbers and their generalizations. A matrix representation for generalized Fibonacci numbers is given, too.

Symmetry of Narayana Numbers and Rowvacuation of Root Posets

For a Weyl group W of rank r, the W-Catalan number is the number of antichains of the poset of positive roots, and the W-Narayana numbers refine the W-Catalan number by keeping track of the cardinalities of these antichains. The W-Narayana numbers are symmetric – that is, the number of antichains of cardinality k is the same as the number of cardinality $r-k$ . However, this symmetry is far from obvious. Panyushev posed the problem of defining an involution on root poset antichains that exhibits the symmetry of the W-Narayana numbers. Rowmotion and rowvacuation are two related operators, defined as compositions of toggles, that give a dihedral action on the set of antichains of any ranked poset. Rowmotion acting on root posets has been the subject of a significant amount of research in the recent past. We prove that for the root posets of classical types, rowvacuation is Panyushev’s desired involution.

Asymptotic normality in t-stack sortable permutations

In this paper, we show that the numbers of t-stack sortable n-permutations with k − 1 descents satisfy central and local limit theorems for t = 1, 2, n − 1 and n − 2. This result, in particular, gives an affirmative answer to Shapiro's question about the asymptotic normality of the Narayana numbers.

Lefschetz Theory for Exterior Algebras and Fermionic Diagonal Coinvariants

Let $W$ be an irreducible complex reflection group acting on its reflection representation $V$. We consider the doubly graded action of $W$ on the exterior algebra $\wedge (V \oplus V^*)$ as well as its quotient $DR_W:= \wedge (V \oplus V^*)/ \langle \wedge (V \oplus V^*)^{W}_+ \rangle $ by the ideal generated by its homogeneous $W$-invariants with vanishing constant term. We describe the bigraded isomorphism type of $DR_W$; when $W = {{\mathfrak{S}}}_n$ is the symmetric group, the answer is a difference of Kronecker products of hook-shaped ${{\mathfrak{S}}}_n$-modules. We relate the Hilbert series of $DR_W$ to the (type A) Catalan and Narayana numbers and describe a standard monomial basis of $DR_W$ using a variant of Motzkin paths. Our methods are type-uniform and involve a Lefschetz-like theory, which applies to the exterior algebra $\wedge (V \oplus V^*)$.