RECORD
Skew doubled shifted plane partitions: calculus and asymptotics
Combinatorics · Submitted 18 Jul 2017
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim In this paper, we establish a new summation formula for Schur processes, called the complete summation formula.
The compiled record
The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
Open the compiled record
The abstract, sentence by sentence
Plane partitions have been widely studied in Mathematics since MacMahon. See, for example, the works by Andrews, Macdonald, Stanley, Sagan and Krattenthaler. The Schur process approach, introduced by Okounkov and Reshetikhin, and further developed by Borodin, Corwin, Corteel, Savelief and Vuleti\'c, has been proved to be a powerful tool in the study of various kinds of plane partitions. The exact enumerations of ordinary plane partitions, shifted plane partitions and cylindric partitions could be derived from two summation formulas for Schur processes, namely, the open summation formula and the cylindric summation formula. In this paper, we establish a new summation formula for Schur processes, called the complete summation formula. As an application, we obtain the generating function and the asymptotic formula for the number of doubled shifted plane partitions, which can be viewed as plane partitions `shifted at the two sides'. We prove that the order of the asymptotic formula depends only on the diagonal width of the doubled shifted plane partition, not on the profile (the skew zone) itself. By using the same methods, the generating function and the asymptotic formula for the number of symmetric cylindric partitions are also derived.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Plane partitions have been widely studied in Mathematics since MacMahon. | background 1.00 | 0.03 | 0.01 | asserted 1.00 |
| s1 | See, for example, the works by Andrews, Macdonald, Stanley, Sagan and Krattenthaler. | background 1.00 | 0.02 | 0.01 | asserted 0.54 |
| s2 | The Schur process approach, introduced by Okounkov and Reshetikhin, and further developed by Borodin, Corwin, Corteel, Savelief and Vuleti\'c, has been proved to be a powerful tool in the study of various kinds of plane partitions. | background 0.99 | 0.04 | 0.02 | asserted 1.00 |
| s3 | The exact enumerations of ordinary plane partitions, shifted plane partitions and cylindric partitions could be derived from two summation formulas for Schur processes, namely, the open summation formula and the cylindric summation formula. | background 1.00 | 0.07 | 0.04 | hedged possibility 0.93 |
| s4 | In this paper, we establish a new summation formula for Schur processes, called the complete summation formula. | contribution 0.98 | 0.97 | 0.49 | asserted 1.00 |
| s5 | As an application, we obtain the generating function and the asymptotic formula for the number of doubled shifted plane partitions, which can be viewed as plane partitions `shifted at the two sides'. | result 0.91 | 0.97 | 0.44 | asserted 1.00 |
| s6 | We prove that the order of the asymptotic formula depends only on the diagonal width of the doubled shifted plane partition, not on the profile (the skew zone) itself. | result 0.99 | 0.95 | 0.28 | asserted 1.00 |
| s7 | By using the same methods, the generating function and the asymptotic formula for the number of symmetric cylindric partitions are also derived. | result 0.81 | 0.94 | 0.08 | asserted 1.00 |
What was examined, and what was not
- arXiv listing abstract v2
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
The paper
Original abstract
Plane partitions have been widely studied in Mathematics since MacMahon. See, for example, the works by Andrews, Macdonald, Stanley, Sagan and Krattenthaler. The Schur process approach, introduced by Okounkov and Reshetikhin, and further developed by Borodin, Corwin, Corteel, Savelief and Vuleti\'c, has been proved to be a powerful tool in the study of various kinds of plane partitions. The exact enumerations of ordinary plane partitions, shifted plane partitions and cylindric partitions could be derived from two summation formulas for Schur processes, namely, the open summation formula and the cylindric summation formula. In this paper, we establish a new summation formula for Schur processes, called the complete summation formula. As an application, we obtain the generating function and the asymptotic formula for the number of doubled shifted plane partitions, which can be viewed as plane partitions `shifted at the two sides'. We prove that the order of the asymptotic formula depends only on the diagonal width of the doubled shifted plane partition, not on the profile (the skew zone) itself. By using the same methods, the generating function and the asymptotic formula for the number of symmetric cylindric partitions are also derived.
Human review
How the literature uses it
1 arXiv paper cite this in 4 sentences
Papers citing it · 3 sentences without a confident label
Uses its method or tool 1 paper
-
For the method of proofs, we will employ a more classical approach based on Mellin transforms (see [ref] ) and saddle-point method (see [ref] ), instead of the elementary approach used in [ref] , which becomes cumbersome when finer asymptotic expansions are required.
Phase transitions from $\exp(n^{1/2})$ to $\exp(n^{2/3})$ in the asymptotics of banded plane partitions · Introduction · 0.98
What it cites · 7 sentences without a confident label
Cites as background 2 papers
-
The Macdonald process, which is a $(q,t)$-generalization of the Schur process, was first introduced by Vuletic [ref] , and further developed by Corteel, Savelief, Vuletić and Langer [ref] in the study of weighted cylindric partitions and plane overpartitions.
Enumeration of Cylindric Plane Partitions - part I · Introduction · 0.88
-
The Macdonald process, which is a $(q,t)$-generalization of the Schur process, was first introduced by Vuletic [ref] , and further developed by Corteel, Savelief, Vuletić and Langer [ref] in the study of weighted cylindric partitions and plane overpartitions.
Enumeration of Cylindric Plane Partitions - Part II · Introduction · 0.88
Discussion
Know something about this paper? Leave a comment, or highlight any passage above to comment on it.