40,350 research outputs found
Jacobi's Identity and Synchronized Partitions
We obtain a finite form of Jacobi's identity and present a combinatorial
proof based on the structure of synchronized partitions.Comment: 7 page
Weighted Forms of Euler's Theorem
In answer to a question of Andrews about finding combinatorial proofs of two
identities in Ramanujan's "Lost" Notebook, we obtain weighted forms of Euler's
theorem on partitions with odd parts and distinct parts. This work is inspired
by the insight of Andrews on the connection between Ramanujan's identities and
Euler's theorem. Our combinatorial formulations of Ramanujan's identities rely
on the notion of rooted partitions. Iterated Dyson's map and Sylvester's
bijection are the main ingredients in the weighted forms of Euler's theorem.Comment: 14 page
k-Marked Dyson Symbols and Congruences for Moments of Cranks
By introducing -marked Durfee symbols, Andrews found a combinatorial
interpretation of -th symmetrized moment of ranks of
partitions of . Recently, Garvan introduced the -th symmetrized moment
of cranks of partitions of in the study of the higher-order
spt-function . In this paper, we give a combinatorial interpretation
of . We introduce -marked Dyson symbols based on a
representation of ordinary partitions given by Dyson, and we show that
equals the number of -marked Dyson symbols of . We then
introduce the full crank of a -marked Dyson symbol and show that there exist
an infinite family of congruences for the full crank function of -marked
Dyson symbols which implies that for fixed prime and positive
integers and , there exist infinitely many non-nested
arithmetic progressions such that .Comment: 19 pages, 2 figure
On the Positive Moments of Ranks of Partitions
By introducing -marked Durfee symbols, Andrews found a combinatorial
interpretation of -th symmetrized moment of ranks of
partitions of in terms of -marked Durfee symbols of . In this
paper, we consider the -th symmetrized positive moment of
ranks of partitions of which is defined as the truncated sum over positive
ranks of partitions of . As combintorial interpretations of
and , we show that for fixed and
with , equals the number of
-marked Durfee symbols of with the -th rank being zero and
equals the number of -marked Durfee symbols of
with the -th rank being positive. The interpretations of
and also imply the interpretation
of given by Andrews since equals
plus twice of . Moreover, we obtain
the generating functions of and .Comment: 10 page
Spin Squeezing of One-Axis Twisting Model in The Presence of Phase Dephasing
We present a detailed analysis of spin squeezing of the one-axis twisting
model with a many-body phase dephasing, which is induced by external field
fluctuation in a two-mode Bose-Einstein condensates. Even in the presence of
the dephasing, our analytical results show that the optimal initial state
corresponds to a coherent spin state with the
polar angle . If the dephasing rate , where
is total atomic spin, we find that the smallest value of squeezing
parameter (i.e., the strongest squeezing) obeys the same scaling with the ideal
one-axis twisting case, namely . While for a moderate
dephasing, the achievable squeezing obeys the power rule , which is
slightly worse than the ideal case. When the dephasing rate ,
we show that the squeezing is weak and neglectable.Comment: 14.2pages, 3 figure
Partition Identities for Ramanujan's Third Order Mock Theta Functions
We find two involutions on partitions that lead to partition identities for
Ramanujan's third order mock theta functions and . We also
give an involution for Fine's partition identity on the mock theta function
f(q). The two classical identities of Ramanujan on third order mock theta
functions are consequences of these partition identities. Our combinatorial
constructions also apply to Andrews' generalizations of Ramanujan's identities.Comment: 12 pages, 1 figur
BG-ranks and 2-cores
We find the number of partitions of whose BG-rank is , in terms of
, the number of pairs of partitions whose total number of cells is ,
giving both bijective and generating function proofs. Next we find congruences
mod 5 for , and then we use these to give a new proof of a refined
system of congruences for that was found by Berkovich and Garvan
Proof of the Andrews-Dyson-Rhoades Conjecture on the spt-Crank
The notion of the spt-crank of a vector partition, or an -partition, was
introduced by Andrews, Garvan and Liang. Let denote the number of
-partitions of with spt-crank . Andrews, Dyson and Rhoades
conjectured that is unimodal for any , and they showed that
this conjecture is equivalent to an inequality between the rank and the crank
of ordinary partitions. They obtained an asymptotic formula for the difference
between the rank and the crank of ordinary partitions, which implies
for sufficiently large and fixed . In this
paper, we introduce a representation of an ordinary partition, called the
-Durfee rectangle symbol, which is a rectangular generalization of the
Durfee symbol introduced by Andrews. We give a proof of the conjecture of
Andrews, Dyson and Rhoades by considering two cases. For , we
construct an injection from the set of ordinary partitions of such that
appears in the rank-set to the set of ordinary partitions of with rank not
less than . The case for requires five more injections. We also show
that this conjecture implies an inequality between the positive rank and crank
moments obtained by Andrews, Chan and Kim.Comment: 34 pages, 2 figure
On Stanley's Partition Function
Stanley defined a partition function t(n) as the number of partitions
of n such that the number of odd parts of is congruent to
the number of odd parts of the conjugate partition modulo 4. We show
that t(n) equals the number of partitions of n with an even number of hooks of
even length. We derive a closed-form formula for the generating function for
the numbers p(n)-t(n). As a consequence, we see that t(n) has the same parity
as the ordinary partition function p(n) for any n. A simple combinatorial
explanation of this fact is also provided.Comment: 8 page
Nearly Equal Distributions of the Rank and the Crank of Partitions
Let denote the number of partitions of with rank not
greater than , and let denote the number of partitions of
with crank not greater than . Bringmann and Mahlburg observed that for and . They
also pointed out that these inequalities can be restated as the existence of a
re-ordering on the set of partitions of such that
or for all
partitions of , that is, the rank and the crank are nearly equal
distributions over partitions of . In the study of the spt-function,
Andrews, Dyson and Rhoades proposed a conjecture on the unimodality of the
spt-crank, and they showed that this conjecture is equivalent to the inequality
for and . We proved this
conjecture by combiantorial arguments. In this paper, we prove the inequality
for and . Furthermore, we define a
re-ordering of the partitions of and show that this
re-ordering leads to the nearly equal distribution of the rank and the
crank. Using the re-ordering , we give a new combinatorial
interpretation of the function ospt defined by Andrews, Chan and Kim,
which immediately leads to an upper bound for due to Chan and Mao.Comment: 19 pages, 1 figur
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