arXiv subjects

Bruce Sagan

Publications and source records attributed to Bruce Sagan.

34 records · Page 2Linked to original sources

arXiv2012

Inversion polynomials for 321-avoiding permutations

We prove a generalization of a conjecture of Dokos, Dwyer, Johnson, Sagan, and Selsor giving a recursion for the inversion polynomial of 321-avoiding permutations. We also answer a question they posed about finding a recursive formulas for the major index polynomial of 321-avoiding permutations. Other properties of these polynomials are investigated as well. Our tools include Dyck and 2-Motzkin paths, polyominoes, and continued fractions.

math.CO↗

arXiv2012

Permutations with given peak set

Let Sym_n denote the symmetric group of all permutations pi = a_1...a_n of {1,...,n}. An index i is a peak of pi if a_{i-1} < a_i > a_{i+1} and we let P(pi) be the set of peaks of pi. Given any set S of positive integers we define P(S;n) to be the set pi in Sym_n with P(pi)=S. Our main result is that for all fixed subsets of positive integers S and all sufficiently large n we have #P(S;n)= p(n) 2^{n-#S-1} for some polynomial p(n) depending on S. We explicitly compute p(n) for various S of probabilistic interest, including certain cases where S depends on n. We also discuss two conjectures, one about positivity of the coefficients of the expansion of p(n) in a binomial coefficient basis, and the other about sets S maximizing #P(S;n) when #S is fixed.

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arXiv2011

Discrete Morse theory and the consecutive pattern poset

We use discrete Morse theory to provide another proof of Bernini, Ferrari, and Steingrimson's formula for the Mobius function of the consecutive pattern poset. In addition, we are able to determine the homotopy type of this poset. Earlier, Bjorner determined the Mobius function and homotopy type of factor order and the results are remarkably similar to those in the pattern case. In his thesis, Willenbring used discrete Morse theory to give an illuminating proof of Bjorner's result. Since our proof parallels Willenbring's, we also consider the relationship between the two posets.

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arXiv2009

Combinatorial interpretations of binomial coefficient analogues related to Lucas sequences

Let s and t be variables. Define polynomials {n} in s, t by {0}=0, {1}=1, and {n}=s{n-1}+t{n-2} for n >= 2. If s, t are integers then the corresponding sequence of integers is called a Lucas sequence. Define an analogue of the binomial coefficients by C{n,k}={n}!/({k}!{n-k}!) where {n}!={1}{2}...{n}. It is easy to see that C{n,k} is a polynomial in s and t. The purpose of this note is to give two combinatorial interpretations for this polynomial in terms of statistics on integer partitions inside a k by n-k rectangle. When s=t=1 we obtain combinatorial interpretations of the fibonomial coefficients which are simpler than any that have previously appeared in the literature.

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arXiv2009

Eulerian quasisymmetric functions and cyclic sieving

It is shown that a refined version of a q-analogue of the Eulerian numbers together with the action, by conjugation, of the subgroup of the symmetric group $S_n$ generated by the $n$-cycle $(1,2,...,n)$ on the set of permutations of fixed cycle type and fixed number of excedances provides an instance of the cyclic sieving phenonmenon of Reiner, Stanton and White. The main tool is a class of symmetric functions recently introduced in work of two of the authors.

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arXiv2008

Set partition statistics and q-Fibonacci numbers

We consider the set partition statistics ls and rb introduced by Wachs and White and investigate their distribution over set partitions avoiding certain patterns. In particular, we consider those set partitions avoiding the pattern 13/2, $Π_n(13/2)$, and those avoiding both 13/2 and 123, $Π_n(13/2,123)$. We show that the distribution over $Π_n(13/2)$ enumerates certain integer partitions, and the distribution over $Π_n(13/2,123)$ gives q-Fibonacci numbers. These q-Fibonacci numbers are closely related to q-Fibonacci numbers studied by Carlitz and by Cigler. We provide combinatorial proofs that these q-Fibonacci numbers satisfy q-analogues of many Fibonacci identities. Finally, we indicate how p,q-Fibonacci numbers arising from the bistatistic (ls, rb) give rise to p,q-analogues of identities.

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arXiv2006

Monomial Bases for Broken Circuit Complexes

Let F be a field and let G be a finite graph with a total ordering on its edge set. Richard Stanley noted that the Stanley-Reisner ring F(G) of the broken circuit complex of G is Cohen-Macaulay. Jason Brown gave an explicit description of a homogeneous system of parameters for F(G) in terms of fundamental cocircuits in G. So F(G) modulo this hsop is a finite dimensional vector space. We conjecture an explicit monomial basis for this vector space in terms of the circuits of G and prove that the conjecture is true for two infinite families of graphs. We also explore an application of these ideas to bounding the number of acyclic orientations of G from above.

math.CO↗

arXiv2006

What power of two divides a weighted Catalan number?

Given a sequence of integers b = (b_0,b_1,b_2,...) one gives a Dyck path P of length 2n the weight wt(P) = b_{h_1} b_{h_2} ... b_{h_n}, where h_i is the height of the ith ascent of P. The corresponding weighted Catalan number is C_n^b = sum_P wt(P), where the sum is over all Dyck paths of length 2n. So, in particular, the ordinary Catalan numbers C_n correspond to b_i = 1 for all i >= 0. Let xi(n) stand for the base two exponent of n, i.e., the largest power of 2 dividing n. We give a condition on b which implies that xi(C_n^b) = xi(C_n). In the special case b_i=(2i+1)^2, this settles a conjecture of Postnikov about the number of plane Morse links. Our proof generalizes the recent combinatorial proof of Deutsch and Sagan of the classical formula for xi(C_n).

math.CO↗

arXiv2005

Counting permutations by congruence class of major index

Consider S_n, the symmetric group on n letters, and let maj pi denote the major index of a permutation pi in S_n. Given positive integers k,l and nonnegative integers i,j, define m_n^{k,l}(i,j) := number of pi in S_n such that maj pi = i (mod k) and maj pi^{-1} = j (mod l). We prove bijectively that if k,l are relatively prime and at most n then m_n^{k,l}(i,j) = n!/(kl) which, surprisingly, does not depend on i and j. Equivalently, if m_n^{k,l}(i,j) is interpreted as the (i,j)-entry of a matrix m_n^{k,l}, then this is a constant matrix under the stated conditions. This bijection is extended to show the more general result that for d at least 1 and k,l relatively prime, the matrix m_n^{kd,ld} admits a block decompostion where each block is the matrix m_n^{d,d}/(kl). We also give an explicit formula for m_n^{n,n} and show that if p is prime then m_{np}^{p,p} has a simple block decomposition. To prove these results, we use the representation theory of the symmetric group and certain restricted shuffles.

math.CO↗

arXiv2005

The Möbius function of the composition poset

We determine the Möbius function of the poset of compositions of an integer. In fact we give two proofs of this formula, one using an involution and one involving discrete Morse theory. The composition poset turns out to be intimately connected with subword order, whose Möbius function was determined by Björner. We show that using a generalization of subword order, we can obtain both Björner's results and our own as special cases.

math.CO↗

arXiv2005

On divisibility of Narayana numbers by primes

Using Kummer's Theorem, we give a necessary and sufficient condition for a Narayana number to be divisible by a given prime. We use this to derive certain properties of the Narayana triangle.

math.CO↗

arXiv2004

Proper partitions of a polygon and k-Catalan numbers

Let P be a polygon whose vertices have been colored (labeled) cyclically with the numbers 1,2,...,c. Motivated by conjectures of Propp, we are led to consider partitions of P into k-gons which are proper in the sense that each k-gon contains all c colors on its vertices. Counting the number of proper partitions involves a generalization of the k-Catalan numbers. We also show that in certain cases, any proper partition can be obtained from another by a sequence of moves called flips.

math.CO↗

arXiv2003

Topological properties of active orders for matroid bases

Las Vergnas introduced several lattice structures on the bases of an ordered matroid M by using their external and internal activities. He also noted that when computing the Moebius function of these lattices, it was often zero, although he had no explanation for that fact. The purpose of this paper is to provide a topological reason for this phenomenon. In particular, we show that the order complex of the external lattice L of M is homotopic to the independence complex of the restriction M^*|T where M^* is the dual of M and T is the top element of L. We then compute some examples showing that this latter complex is often contractible which forces all its homology groups, and thus its Moebius function, to vanish. A theorem of Björner also helps us to calculate the homology of the matroid complex.

math.CO↗

arXiv2001

Pattern frequency sequences and internal zeros

Consider the number of permutations in the symmetric group on n letters that contain c copies of a given pattern. As c varies (with n held fixed) these numbers form a sequence whose properties we study for the monotone patterns and the patterns 1, l, l-1, ..., 2. We show that, except for the patterns 1, 2 and 2, 1 where the sequence is well-known to be log concave, there are infinitely many n where the sequence has internal zeros.

math.CO↗

arXiv2000

Left-modular elements

Left-modularity is a concept that generalizes modularity in lattice theory. In this paper, we give a characterization of left-modular elements and derive two formulae for the characteristic polynomial of a lattice with such an element, one of which generalizes Stanley's Partial Factorization Theorem for a geometric lattice with a modular element. Both formulae provide us with inductive proofs of Blass and Sagan's Total Factorization Theorem for LL lattices. The characteristic polynomials and Mobius functions of non-crossing partition lattices and shuffle posets are computed as examples.

math.CO↗

arXiv1997

Unimodality and the reflection principle

We show how lattice paths and the reflection principle can be used to give easy proofs of unimodality results. In particular, we give a "one-line" combinatorial proof of the unimodality of the binomial coefficients. Other examples include products of binomial coefficients, polynomials related to the Legendre polynomials, and a result connected to a conjecture of Simion.

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