Sufficient conditions are developed, under which the compound Poisson distribution has maximal entropy within a natural class of probability measures on the nonnegative integers. Recently, one of the ...authors O.T. Johnson, Log-concavity and the maximum entropy property of the Poisson distribution, Stochastic Process. Appl., 117(6) (2007) 791–802 used a semigroup approach to show that the Poisson has maximal entropy among all ultra-log-concave distributions with fixed mean. We show via a non-trivial extension of this semigroup approach that the natural analog of the Poisson maximum entropy property remains valid if the compound Poisson distributions under consideration are log-concave, but that it fails in general. A parallel maximum entropy result is established for the family of compound binomial measures. Sufficient conditions for compound distributions to be log-concave are discussed and applications to combinatorics are examined; new bounds are derived on the entropy of the cardinality of a random independent set in a claw-free graph, and a connection is drawn to Mason’s conjecture for matroids. The present results are primarily motivated by the desire to provide an information-theoretic foundation for compound Poisson approximation and associated limit theorems, analogous to the corresponding developments for the central limit theorem and for Poisson approximation. Our results also demonstrate new links between some probabilistic methods and the combinatorial notions of log-concavity and ultra-log-concavity, and they add to the growing body of work exploring the applications of maximum entropy characterizations to problems in discrete mathematics.
In recent years, the log-concavity of the n-th root of a sequence {Snn}n≥1 has been received a lot of attention. Recently, Sun posed the following conjecture in his new book: the sequences {ann}n≥2 ...and {bnn}n≥1 are log-concave, wherean:=1n∑k=0n−1(n−1k)2(n+kk)24k2−1 andbn:=1n3∑k=0n−1(3k2+3k+1)(n−1k)2(n+kk)2. In this paper, two methods, semi-automatic and analytic methods, are used to confirm Sun's conjecture. The semi-automatic method relies on a criterion on the log-concavity of {Snn}n≥1 given by us and a mathematica package due to Hou and Zhang, while the analytic method relies on a result due to Xia.
In this paper, we prove that the sequences of the Stirling numbers of the first and second kind with higher level are both Pólya frequency and log-concave. Then, we show that some polynomials related ...to the above Stirling numbers with higher level are
q
-log-convex or strongly
q
-log-convex. Furthermore, we establish that the linear transformation related to the Stirling numbers of the first kind with level 2 preserves the log-convexity.
We prove that the Poisson distribution maximises entropy in the class of ultra log–concave distributions, extending a result of Harremoës. The proof uses ideas concerning log-concavity, and a ...semigroup action involving adding Poisson variables and thinning. We go on to show that the entropy is a concave function along this semigroup.
Abstract
In the 1970s, Nicolas proved that the partition function
p
(
n
) is log-concave for
$$ n > 25$$
n
>
25
. In Heim et al. (Ann Comb 27(1):87–108, 2023), a precise conjecture on the ...log-concavity for the plane partition function
$${{\textrm{pp}}}(n)$$
pp
(
n
)
for
$$n >11$$
n
>
11
was stated. This was recently proven by Ono, Pujahari, and Rolen. In this paper, we provide a general picture. We associate to double sequences
$$\{g_d(n)\}_{d,n}$$
{
g
d
(
n
)
}
d
,
n
with
$$g_d(1)=1$$
g
d
(
1
)
=
1
and
$$\begin{aligned} 0 \le g_{d}\left( n\right) -n^{d}\le g_{1}\left( n\right) \left( n-1\right) ^{d-1}, \end{aligned}$$
0
≤
g
d
n
-
n
d
≤
g
1
n
n
-
1
d
-
1
,
polynomials
$$\{P_n^{g_d}(x)\}_{d,n}$$
{
P
n
g
d
(
x
)
}
d
,
n
given by
$$\begin{aligned} \sum _{n=0}^{\infty } P_n^{g_d}(x) \, q^n := {{\textrm{exp}}}\left( x \sum _{n=1}^{\infty } g_d(n) \frac{q^n}{n} \right) =\prod _{n=1}^{\infty } \left( 1 - q^n \right) ^{-x f_d(n)}. \end{aligned}$$
∑
n
=
0
∞
P
n
g
d
(
x
)
q
n
:
=
exp
x
∑
n
=
1
∞
g
d
(
n
)
q
n
n
=
∏
n
=
1
∞
1
-
q
n
-
x
f
d
(
n
)
.
We recover
$$ p(n)= P_n^{\sigma _1}(1)$$
p
(
n
)
=
P
n
σ
1
(
1
)
and
$${{\textrm{pp}}}\left( n\right) = P_n^{\sigma _2}(1)$$
pp
n
=
P
n
σ
2
(
1
)
, where
$$\sigma _d (n):= \sum _{\ell \mid n} \ell ^d$$
σ
d
(
n
)
:
=
∑
ℓ
∣
n
ℓ
d
and
$$f_d(n)= n^{d-1}$$
f
d
(
n
)
=
n
d
-
1
. Let
$$n \ge 6$$
n
≥
6
. Then the sequence
$$\{P_n^{\sigma _d}(1)\}_d$$
{
P
n
σ
d
(
1
)
}
d
is log-concave for almost all
d
if and only if
n
is divisible by 3. Let
$${{\textrm{id}}}(n)=n$$
id
(
n
)
=
n
. Then
$$P_n^{{{\textrm{id}}}(x) = \frac{x}{n} L_{n-1}^{(1)}(-x)$$
P
n
id
(
x
)
=
x
n
L
n
-
1
(
1
)
(
-
x
)
, where
$$L_{n}^{\left( \alpha \right) }\left( x\right) $$
L
n
α
x
denotes the
$$\alpha $$
α
-associated Laguerre polynomial. In this paper, we investigate Turán inequalities
$$\begin{aligned} \Delta _{n}^{g_d}(x) := \left( P_n^{g_d}(x) \right) ^2 - P_{n-1}^{g_d}(x) \, P_{n+1}^{g_d}(x) \ge 0. \end{aligned}$$
Δ
n
g
d
(
x
)
:
=
P
n
g
d
(
x
)
2
-
P
n
-
1
g
d
(
x
)
P
n
+
1
g
d
(
x
)
≥
0
.
Let
$$n \ge 6$$
n
≥
6
and
$$0 \le x < 2 - \frac{12}{n+4}$$
0
≤
x
<
2
-
12
n
+
4
. Then
n
is divisible by 3 if and only if
$$\Delta _{n}^{g_d}(x) \ge 0$$
Δ
n
g
d
(
x
)
≥
0
for almost all
d
. Let
$$n \ge 6$$
n
≥
6
and
$$n \not \equiv 2 \pmod {3}$$
n
≢
2
(
mod
3
)
. Then the condition on
x
can be reduced to
$$x \ge 0$$
x
≥
0
. We determine explicit bounds. As an analogue to Nicolas’ result, we have for
$$g_1= {{\textrm{id}}}$$
g
1
=
id
that
$$\Delta _{n}^{{{\textrm{id}}}(x) \ge 0$$
Δ
n
id
(
x
)
≥
0
for all
$$x \ge 0 $$
x
≥
0
and all
n
.
In this paper, we consider a certain class of inequalities for the partition function of the following form:∏i=1Tp(n+si)≥∏i=1Tp(n+ri), which we call multiplicative inequalities. Given a ...multiplicative inequality with the condition that ∑i=1Tsim≠∑i=1Trim for at least one m≥1, we shall construct a unified framework so as to decide whether such a inequality holds or not. As a consequence, we will see that study of such inequalities has manifold applications. For example, one can retrieve log-concavity property, strong log-concavity, and the multiplicative inequality for p(n) considered by Bessenrodt and Ono, to name a few. Furthermore, we obtain an asymptotic expansion for the finite difference of the logarithm of p(n), denoted by (−1)r−1Δrlogp(n), which generalizes a result by Chen, Wang, and Xie.
q-Kaplansky numbers were considered by Chen and Rota. We find that q-Kaplansky numbers are connected to the symmetric differences of Gaussian polynomials introduced by Reiner and Stanton. Based on ...the work of Reiner and Stanton, we establish the unimodality of q-Kaplansky numbers. We also show that q-Kaplansky numbers are the generating functions for the inversion number and the major index of two special kinds of (0,1)-sequences. Furthermore, we show that q-Kaplansky numbers are strongly q-log-concave.
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