MacMahon's Theorem Research Articles

The other dual of MacMahon's theorem on plane partitions

In this paper we introduce a counterpart structure to the shamrocks studied in the paper A dual of Macmahon's theorem on plane partitions by M. Ciucu and C. Krattenthaler (2013) [5], which, just like the latter, can be included at the center of a lattice hexagon on the triangular lattice so that the region obtained from the hexagon by removing it has its number of lozenge tilings given by a simple product formula. The new structure, called a fern, consists of an arbitrary number of equilateral triangles of alternating orientations lined up along a lattice line. The shamrock and the fern seem to be the only such connected structures with this property. It would be interesting to understand the reason for this.

A q-enumeration of lozenge tilings of a hexagon with three dents

MacMahon's classical theorem on boxed plane partitions states that the generating function of the plane partitions fitting in an a×b×c box is equal toHq(a)Hq(b)Hq(c)Hq(a+b+c)Hq(a+b)Hq(b+c)Hq(c+a), where Hq(n):=[0]q!⋅[1]q!…[n−1]q! and [n]q!:=∏i=1n(1+q+q2+…+qi−1). By viewing a boxed plane partition as a lozenge tiling of a semi-regular hexagon, MacMahon's theorem yields a natural q-enumeration of lozenge tilings of the hexagon. However, such q-enumerations do not appear often in the domain of enumeration of lozenge tilings. In this paper, we consider a new q-enumeration of lozenge tilings of a hexagon with three bowtie-shaped regions removed from three non-consecutive sides.The unweighted version of the result generalizes a problem posed by James Propp on enumeration of lozenge tilings of a hexagon of side-lengths 2n, 2n+3, 2n, 2n+3, 2n, 2n+3 (in cyclic order) with the central unit triangles on the (2n+3)-sides removed. Moreover, our result also implies a q-enumeration of boxed plane partitions with certain constraints.

An Extension of MacMahon's Equidistribution Theorem to Ordered Multiset Partitions

A classical result of MacMahon states that inversion number and major index have the same distribution over permutations of a given multiset. In this work, we prove a strengthening of MacMahon's theorem originally conjectured by Haglund. Our result can be seen as an equidistribution theorem over the ordered partitions of a multiset into sets, which we call ordered multiset partitions. Our proof is bijective and involves a new generalization of Carlitz's insertion method. This generalization leads to a new extension of Macdonald polynomials for hook shapes. We use our main theorem to show that these polynomials are symmetric and we give their Schur expansion.
 
 A corrigendum was added 17 September 2019.

MacMahon's Theorem for a Set of Permutations with Given Descent Indices and Right-Maximal Records

We show that the major codes and inversion codes are equidistributed over a set of permutations with prescribed descent indices and right-maximal records.

Euler–Mahonian statistics on ordered set partitions (II)

[E. Steingrímsson, Statistics on ordered partitions of sets, arXiv: math.CO/0605670] introduced several hard statistics on ordered set partitions and conjectured that their generating functions are related to the q-Stirling numbers of the second kind. In a previous paper, half of these conjectures have been proved by Ishikawa, Kasraoui and Zeng using the transfer-matrix method. In this paper, we shall give bijective proofs of all the conjectures of Steingrímsson. Our basic idea is to encode ordered set partitions by a kind of path diagrams and explore the rich combinatorial properties of the latter structure. As a bonus of our approach, we derive two new σ-partition interpretations of the p , q -Stirling numbers of the second kind introduced by Wachs and White. We also discuss the connections with MacMahon's theorem on the equidistribution of the inversion number and major index on words and give a partition version of his result.

Integrals, partitions and MacMahon's Theorem

In two previous papers, the study of partitions with short sequences has been developed both for its intrinsic interest and for a variety of applications. The object of this paper is to extend that study in various ways. First, the relationship of partitions with no consecutive integers to a theorem of MacMahon and mock theta functions is explored independently. Secondly, we derive in a succinct manner a relevant definite integral related to the asymptotic enumeration of partitions with short sequences. Finally, we provide the generating function for partitions with no sequences of length K and part exceeding N.

Permutation statistics on the alternating group

Let A n ⊆ S n denote the alternating and the symmetric groups on 1 , … , n . MacMahon's theorem [P.A. MacMahon, Combinatory Analysis I–II, Cambridge Univ. Press, 1916], about the equi-distribution of the length and the major indices in S n , has received far reaching refinements and generalizations, by Foata [Proc. Amer. Math. Soc. 19 (1968) 236], Carlitz [Trans. Amer. Math. Soc. 76 (1954) 332; Amer. Math. Monthly 82 (1975) 51], Foata-Schützenberger [Math. Nachr. 83 (1978) 143], Garsia–Gessel [Adv. Math. 31 (1979) 288] and followers. Our main goal is to find analogous statistics and identities for the alternating group A n . A new statistics for S n , the delent number, is introduced. This new statistics is involved with new S n identities, refining some of the results in [D. Foata, M.P. Schützenberger, Math. Nachr. 83 (1978) 143; A.M. Garsia, I. Gessel, Adv. Math. 31 (1979) 288]. By a certain covering map f : A n + 1 → S n , such S n identities are ‘lifted’ to A n + 1 , yielding the corresponding A n + 1 equi-distribution identities.

Notes on plane partitions. I

In this paper an expression is obtained for the generating function B k ( x ) = ∑ n = 0 ∞ b k ( n ) x n where b k ( n) is the number of k-rowed partitions of n whose nonzero parts decrease strictly along each row. As an application, a new proof is given of Mac Mahon's theorem that A k ( x ) = ∑ n = 0 ∞ a k ( n ) x n = ∏ v = 0 ∞ ( 1 − x v ) − min ⁡ ( v , k ) where a k ( n) is the total number of k-rowed partitions of n.