Annals of Combinatorics

Alon, Noga; Butler, Steve; Graham, Ron; Rajkumar, Utkrisht

doi: 10.1007/s00026-018-0403-3pmid: N/A

Let $$M = (s_1, s_2, \ldots , s_n) $$ M = ( s 1 , s 2 , … , s n ) be a sequence of distinct symbols and $$\sigma $$ σ a permutation of $$\{1,2, \ldots , n\}$$ { 1 , 2 , … , n } . Denote by $$\sigma (M)$$ σ ( M ) the permuted sequence $$(s_{\sigma (1)}, s_{\sigma (2)}, \ldots , s_{\sigma (n)})$$ ( s σ ( 1 ) , s σ ( 2 ) , … , s σ ( n ) ) . For a given positive integer d, we will say that $$\sigma $$ σ is d -resilient if no matter how d entries of M are removed from M to form $$M'$$ M ′ and d entries of $$\sigma (M)$$ σ ( M ) are removed from $$\sigma (M)$$ σ ( M ) to form $$\sigma (M)'$$ σ ( M ) ′ (with no symbol being removed from both sequences), it is always possible to reconstruct the original sequence M from $$M'$$ M ′ and $$\sigma (M)'$$ σ ( M ) ′ . Necessary and sufficient conditions for a permutation to be d-resilient are established in terms of whether certain auxiliary graphs are acyclic. We show that for d-resilient permutations for [n] to exist, n must have size at least exponential in d, and we give an algorithm to construct such permutations in this case. We show that for each d and all sufficiently large n, the fraction of all permutations on n elements which are d-resilient is bounded away from 0.

Cheng, Raymond; Jackson, David; Stanley, Geoff

doi: 10.1007/s00026-018-0404-2pmid: N/A

Quasi-triangular Hopf algebras were introduced by Drinfel’d in his construction of solutions to the Yang–Baxter Equation. This algebra is built upon $$\mathscr {U}_h(\mathfrak {sl}_2)$$ U h ( sl 2 ) , the quantized universal enveloping algebra of the Lie algebra $$\mathfrak {sl}_2$$ sl 2 . In this paper, combinatorial structure in $$\mathscr {U}_h(\mathfrak {sl}_2)$$ U h ( sl 2 ) is elicited, and used to assist in highly intricate calculations in this algebra. To this end, a combinatorial methodology is formulated for straightening algebraic expressions to a canonical form in the case $$n=1$$ n = 1 . We apply this formalism to the quasi-triangular Hopf algebras and obtain a constructive account not only for the derivation of the Drinfel’d’s $$sR$$ sR -matrix, but also for the arguably mysterious ribbon elements of $$\mathscr {U}_h(\mathfrak {sl}_2)$$ U h ( sl 2 ) . Finally, we extend these techniques to the higher-dimensional algebras $$\mathscr {U}_h(\mathfrak {sl}_{n+1})$$ U h ( sl n + 1 ) . While these explicit algebraic results are well known, our contribution is in our formalism and perspective: our emphasis is on the combinatorial structure of these algebras and how that structure may guide algebraic constructions.

Cook, Brian; Magyar, Ákos; Titichetrakun, Tatchai

doi: 10.1007/s00026-018-0402-4pmid: N/A

Let A be a subset of positive relative upper density of $$\mathbb {P}^d$$ P d , the d-tuples of primes. We present an essentially self-contained, combinatorial argument to show that A contains infinitely many affine copies of any finite set $$F\subseteq \mathbb {Z}^d$$ F ⊆ Z d . This provides a natural multidimensional extension of the theorem of Green and Tao on the existence of long arithmetic progressions in the primes.

Ehrenborg, Richard

doi: 10.1007/s00026-018-0412-2pmid: N/A

We study how the shelling components behave under the pyramid and prism operations. As a consequence we obtain a concise recursion for the cubical shelling contributions.

Gaudet, Louis; Jensen, David; Ranganathan, Dhruv; Wawrykow, Nicholas; Weisman, Theodore

doi: 10.1007/s00026-018-0406-0pmid: N/A

We study which groups with pairing can occur as the Jacobian of a finite graph. We provide explicit constructions of graphs whose Jacobian realizes a large fraction of odd groups with a given pairing. Conditional on the generalized Riemann hypothesis, these constructions yield all groups with pairing of odd order, and unconditionally, they yield all groups with pairing whose prime factors are sufficiently large. For groups with pairing of even order, we provide a partial answer to this question, for a certain restricted class of pairings. Finally, we explore which finite abelian groups occur as the Jacobian of a simple graph. There exist infinite families of finite abelian groups that do not occur as the Jacobians of simple graphs.

Kim, Byungchan; Toh, Pee

doi: 10.1007/s00026-018-0407-zpmid: N/A

We study a crank function M(m, n) for cubic partition pairs. We show that the function M(m, n) explains a cubic partition pair congruence and we also obtain various arithmetic properties regarding M(m, n). In particular, using the $$\Theta $$ Θ -operator, we confirm a conjecture on the sign pattern of c(n), the number of cubic partition pairs of n, weighted by the parity of the crank.

Reiner, Victor; Sommers, Eric

doi: 10.1007/s00026-018-0408-ypmid: N/A

Catalan numbers are known to count noncrossing set partitions, while Narayana and Kreweras numbers refine this count according to the number of blocks in the set partition, and by its collection of block sizes. Motivated by reflection group generalizations of Catalan numbers and their q-analogues, this paper concerns a definition of q-Kreweras numbers for finite Weyl groups W, refining the q-Catalan numbers for W, and arising from work of the second author. We give explicit formulas in all types for the q-Kreweras numbers. In the classical types A, B, C, we also record formulas for the q-Narayana numbers and in the process show that the formulas depend only on the Weyl group (that is, they coincide in types B and C). In addition, we verify that in the classical types A, B, C, D the q-Kreweras numbers obey the expected cyclic sieving phenomena when evaluated at appropriate roots of unity.

Tsuchiya, Akiyoshi

doi: 10.1007/s00026-018-0405-1pmid: N/A

Let $$(P,\le _P)$$ ( P , ≤ P ) and $$(Q,\le _Q)$$ ( Q , ≤ Q ) be finite partially ordered sets with $$|P|=|Q|=d$$ | P | = | Q | = d , and $$\mathcal {C}(P) \subset \mathbb {R}^d$$ C ( P ) ⊂ R d and $$\mathcal {C}(Q) \subset \mathbb {R}^d$$ C ( Q ) ⊂ R d their chain polytopes. The twinned chain polytope of P and Q is the lattice polytope $$\Gamma (\mathcal {C}(P),\mathcal {C}(Q)) \subset \mathbb {R}^d$$ Γ ( C ( P ) , C ( Q ) ) ⊂ R d which is the convex hull of $$\mathcal {C}(P) \cup (-\mathcal {C}(Q))$$ C ( P ) ∪ ( - C ( Q ) ) . It is known that twinned chain polytopes are Gorenstein Fano polytopes with the integer decomposition property. In the present paper, we study combinatorial properties of twinned chain polytopes. First, we will give the formula of the volume of twinned chain polytopes in terms of the underlying partially ordered sets. Second, we will identify the facet-supporting hyperplanes of twinned chain polytopes in terms of the underlying partially ordered sets. Finally, we will provide the vertex representations of the dual polytopes of twinned chain polytopes.

Wildon, Mark

doi: 10.1007/s00026-018-0409-xpmid: N/A

This paper proves a combinatorial rule expressing the product $$s_\tau (s_{\lambda /\mu } \circ p_r)$$ s τ ( s λ / μ ∘ p r ) of a Schur function and the plethysm of a skew Schur function with a power-sum symmetric function as an integral linear combination of Schur functions. This generalizes the SXP rule for the plethysm $$s_\lambda \circ p_r$$ s λ ∘ p r . Each step in the proof uses either an explicit bijection or a sign-reversing involution. The proof is inspired by an earlier proof of the SXP rule due to Remmel and Shimozono (Discrete Math. 193:257–266, 1998). The connections with two later combinatorial rules for special cases of this plethysm are discussed. Two open problems are raised. The paper is intended to be readable by non-experts.