Annals of Combinatorics最新文献

Discrete Morse theory, a cell complex-analog to smooth Morse theory allowing homotopic tools in the discrete realm, has been developed over the past few decades since its original formulation by Robin Forman in 1998. In particular, discrete gradient vector fields on simplicial complexes capture important topological features of the structure. We prove that the characteristic polynomials of the Laplacian matrices of a simplicial complex are generating functions for discrete gradient vector fields if the complex is a triangulation of an orientable manifold. Furthermore, we provide a full characterization of the correspondence between rooted forests in higher dimensions and discrete gradient vector fields.

This work highlights the existence of partial symmetries in large families of iterated plethystic coefficients. The plethystic coefficients involved come from the expansion in the Schur basis of iterated plethysms of Schur functions indexed by one-row partitions.The partial symmetries are described in terms of an involution on partitions, the flip involution, that generalizes the ubiquitous (omega ) involution. Schur-positive symmetric functions possessing this partial symmetry are termed flip-symmetric. The operation of taking plethysm with (s_lambda ) preserves flip-symmetry, provided that (lambda ) is a partition of two. Explicit formulas for the iterated plethysms (s_2circ s_bcirc s_a) and (s_ccirc s_2circ s_a), with a,  b,  and c (ge ) 2 allow us to show that these two families of iterated plethysms are flip-symmetric. The article concludes with some observations, remarks, and open questions on the unimodality and asymptotic normality of certain flip-symmetric sequences of iterated plethystic coefficients.

Rectangular standard Young tableaux with 2 or 3 rows are in bijection with (U_q(mathfrak {sl}_2))-webs and (U_q(mathfrak {sl}_3))-webs, respectively. When (mathcal {W}) is a web with a reflection symmetry, the corresponding tableau (T_mathcal {W}) has a rotational symmetry. Folding (T_mathcal {W}) transforms it into a domino tableau (D_mathcal {W}). We study the relationships between these correspondences. For 2-row tableaux, folding a rotationally symmetric tableau corresponds to “literally folding” the web along its axis of symmetry. For 3-row tableaux, we give simple algorithms, which provide direct bijective maps between symmetrical webs and domino tableaux (in both directions). These details of these algorithms reflect the intuitive idea that (D_mathcal {W}) corresponds to “(mathcal {W}) modulo symmetry”.

Twins in a finite word are formed by a pair of identical subwords placed at disjoint sets of positions. We investigate the maximum length of twins in a random word over a k-letter alphabet. The obtained lower bounds for small values of k significantly improve the best estimates known in the deterministic case. Bukh and Zhou in 2016 showed that every ternary word of length n contains twins of length at least 0.34n. Our main result states that in a random ternary word of length n, with high probability, one can find twins of length at least 0.41n. In the general case of alphabets of size (kgeqslant 3) we obtain analogous lower bounds of the form (frac{1.64}{k+1}n) which are better than the known deterministic bounds for (kleqslant 354). In addition, we present similar results for multiple twins in random words.

The polytopes (mathcal {U}_{I,overline{J}}) were introduced by Ceballos, Padrol, and Sarmiento to provide a geometric approach to the study of ((I,overline{J}))-Tamari lattices. They observed a connection between certain (mathcal {U}_{I,overline{J}}) and acyclic root polytopes, and wondered if Mészáros’ subdivision algebra can be used to subdivide all (mathcal {U}_{I,overline{J}}). We answer this in the affirmative from two perspectives, one using flow polytopes and the other using root polytopes. We show that (mathcal {U}_{I,overline{J}}) is integrally equivalent to a flow polytope that can be subdivided using the subdivision algebra. Alternatively, we find a suitable projection of (mathcal {U}_{I,overline{J}}) to an acyclic root polytope which allows subdivisions of the root polytope to be lifted back to (mathcal {U}_{I,overline{J}}). As a consequence, this implies that subdivisions of (mathcal {U}_{I,overline{J}}) can be obtained with the algebraic interpretation of using reduced forms of monomials in the subdivision algebra. In addition, we show that the ((I,overline{J}))-Tamari complex can be obtained as a triangulated flow polytope.

A k-plane tree is a plane tree whose vertices are assigned labels between 1 and k in such a way that the sum of the labels along any edge is no greater than (k+1). These trees are known to be related to ((k+1))-ary trees, and they are counted by a generalised version of the Catalan numbers. We prove a surprisingly simple refined counting formula, where we count trees with a prescribed number of labels of each kind. Several corollaries are derived from this formula, and an analogous theorem is proven for k-noncrossing trees, a similarly defined family of labelled noncrossing trees that are related to ((2k+1))-ary trees.

We identify the asymptotic distribution of p-rank of the sandpile group of random directed bipartite graphs which are not too imbalanced. We show this matches exactly with that of the Erdös–Rényi random directed graph model, suggesting that the Sylow p-subgroups of this model may also be Cohen–Lenstra distributed. Our work builds on the results of Koplewitz who studied p-rank distributions for unbalanced random bipartite graphs, and showed that for sufficiently unbalanced graphs, the distribution of p-rank differs from the Cohen–Lenstra distribution. Koplewitz (sandpile groups of random bipartite graphs, https://arxiv.org/abs/1705.07519, 2017) conjectured that for random balanced bipartite graphs, the expected value of p-rank is O(1) for any p. This work proves his conjecture and gives the exact distribution for the subclass of directed graphs.

In 2022, Defant and Kravitz introduced extended promotion (denoted ( partial )), a map that acts on the set of labelings of a poset. Extended promotion is a generalization of Schützenberger’s promotion operator, a well-studied map that permutes the set of linear extensions of a poset. It is known that if L is a labeling of an n-element poset P, then ( partial ^{n-1}(L) ) is a linear extension. This allows us to regard ( partial ) as a sorting operator on the set of all labelings of P, where we think of the linear extensions of P as the labelings which have been sorted. The labelings requiring ( n-1 ) applications of ( partial ) to be sorted are called tangled; the labelings requiring ( n-2 ) applications are called quasi-tangled. We count the quasi-tangled labelings of a relatively large class of posets called inflated rooted trees with deflated leaves. Given an n-element poset with a unique minimal element with the property that the minimal element has exactly one parent, it follows from the aforementioned enumeration that this poset has ( 2(n-1)!-(n-2)! ) quasi-tangled labelings. Using similar methods, we outline an algorithmic approach to enumerating the labelings requiring ( n-k-1 ) applications to be sorted for any fixed ( kin {1,ldots ,n-2} ). We also make partial progress towards proving a conjecture of Defant and Kravitz on the maximum possible number of tangled labelings of an n-element poset.

For a positive integer (tge 2), let (b_{t}(n)) denote the number of t-regular partitions of a nonnegative integer n. Motivated by some recent conjectures of Keith and Zanello, we establish infinite families of congruences modulo 2 for (b_9(n)) and (b_{19}(n)). We prove some specific cases of two conjectures of Keith and Zanello on self-similarities of (b_9(n)) and (b_{19}(n)) modulo 2. For (tin {6,10,14,15,18,20,22,26,27,28}), Keith and Zanello conjectured that there are no integers (A>0) and (Bge 0) for which (b_t(An+ B)equiv 0pmod 2) for all (nge 0). We prove that, for any (tge 2) and prime (ell ), there are infinitely many arithmetic progressions (An+B) for which (sum _{n=0}^{infty }b_t(An+B)q^nnot equiv 0 pmod {ell }). Next, we obtain quantitative estimates for the distributions of (b_{6}(n), b_{10}(n)) and (b_{14}(n)) modulo 2. We further study the odd densities of certain infinite families of eta-quotients related to the 7-regular and 13-regular partition functions.