A sprout sequence is a sequence =(R_0=1,R_1,R_2,…) of symmetric functions in the variables =(x_1,x_2,…) over a field K generated from a power series F(t)=1+a_1t+a_2t^2+⋯ by the rule ∑_n≥ 0R_nt^n = ∏_i≥ 1 F(x_it). The power series F(t) is called the seed of . This concept originated in the work of Littlewood and Richardson (though not with the name “sprout sequence”), and numerous examples of sprout sequences have appeared in the literature. They are related to chromatic Tutte polynomials of complete graphs and complete hypergraphs, binomial posets, upper homogeneous (upho) posets, topological genera, etc. We first develop the basic theory of sprout sequences and then look at the special case F(t)=(√(t)). We give five characterizations of sprout sequences and consider the expansion of sprout symmetric functions in terms of well-known symmetric function bases. The Schur positivity, elementary symmetric function positivity, and complete homogeneous symmetric function positivity of R_n for all n are completely characterized using the Edrei-Thoma theorem from the theory of total positivity. The seed F(t)=(√(t)) is especially interesting. The expansion of R_n in the power sum or monomial basis is related to alternating permutations. The Schur function expansion is related to standard Young skew tableaux. The expansion in terms of the complete symmetric functions has nonnegative integer coefficients, but we don't know a combinatorial interpretation. Finally we give a formula for R_n as a sum of chromatic symmetric functions of interval orders.

A parking function is a sequence (er1, ... , ern) of positive integers such that if )1 & centerdot; & centerdot; & centerdot; )n is the increasing rearrangement of er1, ... , ern, then )i i for 1 i n. In this paper we obtain some new results on the enumeration of parking functions. We will consider the joint distribution of several sets of statistics on parking functions. The distribution of most of these individual statistics is known, but the joint distributions are new. Parking functions of length n are in bijection with labelled forests on the vertex set [n] = {1, 2,.. . , n} (or rooted trees on [n]0 = {0, 1,.. . , n} with root 0), so our results can also be applied to labelled forests. Extensions of our techniques are discussed, including an extension to a probabilistic scenario.

Let Fi denote the ith Fibonacci number, and define ∏i=1n1+xFi+1=∑kcn(k)xk. The paper is concerned primarily with the coefficients cn(k). In particular, for any r≥0 the generating function ∑n≥0(∑kcn(k)r)xn is rational. The coefficients cn(k) can be displayed in an array called the Fibonacci triangle poset F with some interesting further properties, including an encoding of a certain dense linear order on the nonnegative integers. Some generalizations are briefly considered, but there remain many open questions.

We define a "shifted analogue" SH_n of the parking function symmetric function PF_n. The expansion of SH_n in terms of three bases for shifted symmetric functions is explicitly described. We don't know a shifted analogue for parking functions themselves, but some desirable properties of such an analogue are discussed.

A parking function is a sequence $(a_1,\dots, a_n)$ of positive integers such that if $b_1\leq\cdots\leq b_n$ is the increasing rearrangement of $a_1,\dots,a_n$, then $b_i\leq i$ for $1\leq i\leq n$. In this paper we obtain some new results on the enumeration of parking functions. We will consider the joint distribution of several sets of statistics on parking functions. The distribution of most of these individual statistics is known, but the joint distributions are new. Parking functions of length $n$ are in bijection with labelled forests on the vertex set $[n]=\{1,2,\dots,n\}$ (or rooted trees on $[n]_0=\{0,1,\dots,n\}$ with root $0$), so our results can also be applied to labelled forests. Extensions of our techniques are discussed.

We consider a linear operator ψ r from the ring Λ Q of symmetric functions over Q to the polynomial ringwhere m λ is a monomial symmetric function, (λ i ) r denotes the falling factorial, andWe obtain formulas for many instances of ψ r b λ , where b λ denotes one of the six standard bases for Λ Q .The formula for ψ 2 s λ , where s λ is a Schur function, is equivalent to a formula of M. Thiel and N. Williams on the expected square norm of the weight of an irreducible representation of the Lie algebra sl(n, C).

Let $D=\left( V,A\right) $ be a digraph with $n$ vertices, where each arc $a\in A$ is a pair $\left( u,v\right) $ of two vertices. We study the \emph{Redei--Berge symmetric function} $U_{D}$, defined as the quasisymmetric function% \[ \sum L_{\operatorname*{Des}\left( w,D\right) ,\ n}\in\operatorname*{QSym}. \] Here, the sum ranges over all lists $w=\left( w_{1},w_{2},\ldots ,w_{n}\right) $ that contain each vertex of $D$ exactly once, and the corresponding addend is% \[ L_{\operatorname*{Des}\left( w,D\right) ,\ n}:=\sum_{\substack{i_{1}\leq i_{2}\leq\cdots\leq i_{n};\\i_{p}<i_{p+1}\text{ for each }p\text{ satisfying }\left( w_{p},w_{p+1}\right) \in A}}x_{i_{1}}x_{i_{2}}\cdots x_{i_{n}}% \] (an instance of Gessel's fundamental quasisymmetric functions). While $U_{D}$ is a specialization of Chow's path-cycle symmetric function, which has been studied before, we prove some new formulas that express $U_{D}$ in terms of the power-sum symmetric functions. We show that $U_{D}$ is always $p$-integral, and furthermore is $p$-positive whenever $D$ has no $2$-cycles. When $D$ is a tournament, $U_{D}$ can be written as a polynomial in $p_{1},2p_{3},2p_{5},2p_{7},\ldots$ with nonnegative integer coefficients. By specializing these results, we obtain the famous theorems of Redei and Berge on the number of Hamiltonian paths in digraphs and tournaments, as well as a modulo-$4$ refinement of Redei's theorem.

Let $P$ be a finite poset of width two, i.e., with no three-element antichain. We associate with $P$ a skew Young diagram $\Upsilon(P)$ and discuss some of the properties of the map $\Upsilon$. In particular, if we regard $\Upsilon(P)$ as a poset in a standard way, then the linear extensions of $P$ are in bijection with the order ideals of $\Upsilon(P)$.

We initiate a study of the representation of the symmetric group on the multilinear component of an n-ary generalization of the free Lie algebra, which we call a free LAnKe. Our central result is that the representation of the symmetric group S2n−1 on the multilinear component of the free LAnKe with 2n−1 generators is given by an irreducible representation whose dimension is the nth Catalan number. This leads to a more general result on eigenspaces of a certain linear operator, which has additional consequences. We also obtain a new presentation of Specht modules of staircase shape as a consequence of our central result.

A partly autobiographical survey of the development of enumerative and algebraic combinatorics in the 1960's and 1970's.

We define a triangular array closely related to Stern's diatomic array and show that for a fixed integer r≥ 1, the sum u_r(n) of the rth powers of the entries in row n satisfy a linear recurrence with constant coefficients. The proof technique yields a vast generalization. In certain cases we can be more explicit about the resulting linear recurrence.

An interesting, and still wide open, conjecture of Reiner and Stanton predicts that certain strange symmetric differences of $q$-binomial coefficients are always nonnegative and unimodal. We extend their conjecture to a broader, and perhaps more natural, framework, by conjecturing that, for each $k\ge 5$, the polynomials $$f(k,m,b)(q)=\binom{m}{k}_q-q^{\frac{k(m-b)}{2}+b-2k+2}\cdot\binom{b}{k-2}_q$$ are nonnegative and unimodal for all $m\gg_k 0$ and $b\le \frac{km-4k+4}{k-2}$ such that $kb\equiv km$ (mod 2), with the only exception of $b=\frac{km-4k+2}{k-2}$ when this is an integer. Using the KOH theorem, we combinatorially show the case $k=5$. In fact, we completely characterize the nonnegativity and unimodality of $f(k,m,b)$ for $k\le 5$. (This also provides an isolated counterexample to Reiner-Stanton's conjecture when $k=3$.) Further, we prove that, for each $k$ and $m$, it suffices to show our conjecture for the largest $2k-6$ values of $b$.

Roughly ten years ago, the following "Gorenstein Interval Conjecture" (GIC) was proposed: Whenever (1, h(1), ..., h(i), ..., h(e-i), ..., h(c-1), 1) and (1, h(1), ..., h(i )+ alpha, ..., h(e-i) + alpha, ..., h(e-1), 1) are both Gorenstein Hilbert functions for some alpha >= 2, then (1, h(1), ..., h(i), + beta, ..., h(e-i) + beta, ..., h(e-1), 1) is also Gorenstein, for all beta = 1,2, ..., alpha - 1. Since an explicit characterization of which Hilbert functions are Gorenstein is widely believed to be hopeless, the GIC, if true, would at least provide the existence of a strong, and very natural, structural property for such basic functions in commutative algebra. Before now, very little progress was made on the GIC. The main goal of this note is to prove the case e <= 5, in arbitrary codimension. Our arguments will be in part constructive, and will combine several different tools of commutative algebra and classical algebraic geometry. (C) 2019 Elsevier Inc. All rights reserved.

We look at the number L(n) of O-sequences of length n. Recall that an O-sequence can be defined algebraically as the Hilbert function of a standard graded k-algebra, or combinatorially as the f-vector of a multicomplex. The sequence L(n) was first investigated in a recent paper by commutative algebraists Enkosky and Stone, inspired by Huneke. In this note, we significantly improve both of their upper and lower bounds, by means of a very short partition-theoretic argument. In particular, it turns out that, for suitable positive constants c1 and c2 and all n>2, ec1n≤L(n)≤ec2nlogn. It remains an open problem to determine an exact asymptotic estimate for L(n).

While much research has been done on the Ehrhart functions of integral and rational polytopes, little is known in the irrational case. In our main theorem, we determine exactly when the Ehrhart function of a right triangle with legs on the axes and slant edge with irrational slope is a polynomial. We also investigate several other situations where the period of the Ehrhart function of a polytope is less than the denominator of that polytope. For example, we give examples of irrational polytopes with polynomial Ehrhart function in any dimension, and we find triangles with periods dividing any even-index k-Fibonacci number, but with larger denominators.

An evil warden is in charge of 100 prisoners (all with different names). He puts a row of 100 boxes in a room. Inside each box is the name of a different prisoner. The prisoners enter the room one at a time. Each prisoner must open 50 of the boxes, one at a time. If any of the prisoners does not see his or her own name, then they are all killed. The prisoners may have a discussion before the first prisoner enters the room with the boxes, but after that there is no further communication. A prisoner may not leave a message of any kind for another prisoner. In particular, all the boxes are shut once a prisoner leaves the room. If all the prisoners choose 50 boxes at random, then each has a success probability of 1/2, so the probability that they are not killed is 2−100, not such good odds. Is there a strategy that will increase the chances of success? What is the best strategy?

In Chapters 5 and 6 we considered the quotient poset B n ∕G, where G is a subgroup of the symmetric group $$\mathfrak {S}_n$$ . If p i is the number of elements of rank i of this poset, then the sequence p 0, p 1, …, p n is rank-symmetric and rank-unimodal.