Plane Partitions as Sums over Partitions

Merca, Mircea; Radu, Iulia-Ionelia

doi:10.3390/sym15101820

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2023

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Cited by 3 publications

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“…Recently, Merca and Radu [6] considered specializations of complete homogeneous symmetric functions and provided the following formula for PL (m,r) (n).…”

Section: Sincementioning

confidence: 99%

Plane Partitions and Divisors

Ballantine,

Merca

2023

Symmetry

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“…Recently, Merca and Radu [6] considered specializations of complete homogeneous symmetric functions and provided the following formula for PL (m,r) (n).…”

Section: Sincementioning

confidence: 99%

Plane Partitions and Divisors

Ballantine,

Merca

2023

Symmetry

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Plane Partitions and a Problem of Josephus

Merca

2023

Mathematics

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The Josephus Problem is a mathematical counting-out problem with a grim description: given a group of n persons arranged in a circle under the edict that every kth person will be executed going around the circle until only one remains, find the position L(n,k) in which you should stand in order to be the last survivor. Let Jn be the order in which the first person is executed on counting when k=2. In this paper, we consider the sequence (Jn)n⩾1 in order to introduce new expressions for the generating functions of the number of strict plane partitions and the number of symmetric plane partitions. This approach allows us to express the number of strict plane partitions of n and the number of symmetric plane partitions of n as sums over partitions of n in terms of binomial coefficients involving Jn. Also, we introduce interpretations for the strict plane partitions and the symmetric plane partitions in terms of colored partitions. Connections between the sum of the divisors’ functions and Jn are provided in this context.

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n-Color Partitions into Distinct Parts as Sums over Partitions

Merca,

Simion

2023

Symmetry

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The partitions in which the parts of size n can come in n different colors are known as n-color partitions. For r∈{0,1}, let QLr(n) be the number of n-color partitions of n into distinct parts which have a number of parts congruent to r modulo 2. In this paper, we consider specializations of complete and elementary symmetric functions in order to establish two kinds of formulas for QL0(n)±QL1(n) as sums over partitions of n in terms of binomial coefficients. The first kind of formulas only involve partitions in which the parts of size n appear at most n times, while the second kind of formulas involve unrestricted partitions. Similar results are obtained for the first differences of QL0(n)±QL1(n) and the partial sums of QL0(n)±QL1(n).

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Plane Partitions as Sums over Partitions

Cited by 3 publications

References 6 publications

Plane Partitions and Divisors

Plane Partitions and Divisors

Plane Partitions and a Problem of Josephus

n-Color Partitions into Distinct Parts as Sums over Partitions

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