67 research outputs found
Central and local limit theorems for the coefficients of polynomials of binomial type
AbstractWe introduce the problem of establishing a central limit theorem for the coefficients of a sequence of polynomials Pn(x) of binomial type; that is, a sequence Pn(x) satisfying exp(xg(u)) = βn=0β Pn(x)(unn!) for some (formal) power series g(u) lacking constant term. We give a complete answer in the case when g(u) is a polynomial, and point out the widest known class of nonpolynomial power series g(u) for which the corresponding central limit theorem is known true. We also give the least restrictive conditions known for the coefficients of Pn(x) which permit passage from a central to a local limit theorem, as well as a simple criterion for the generating function g(u) which assures these conditions on the coefficients of Pn(x). The latter criterion is a new and general result concerning log concavity of doubly indexed sequences of numbers with combinatorial significance. Asymptotic formulas for the coefficients of Pn(x) are developed
Asymptotic enumeration of dense 0-1 matrices with specified line sums
Let S=(s_1,s_2,..., s_m) and T = (t_1,t_2,..., t_n) be vectors of
non-negative integers with sum_{i=1}^{m} s_i = sum_{j=1}^n t_j. Let B(S,T) be
the number of m*n matrices over {0,1} with j-th row sum equal to s_j for 1 <= j
<= m and k-th column sum equal to t_k for 1 <= k <= n. Equivalently, B(S,T) is
the number of bipartite graphs with m vertices in one part with degrees given
by S, and n vertices in the other part with degrees given by T. Most research
on the asymptotics of B(S,T) has focused on the sparse case, where the best
result is that of Greenhill, McKay and Wang (2006). In the case of dense
matrices, the only precise result is for the case of equal row sums and equal
column sums (Canfield and McKay, 2005). This paper extends the analytic methods
used by the latter paper to the case where the row and column sums can vary
within certain limits. Interestingly, the result can be expressed by the same
formula which holds in the sparse case.Comment: Multiple minor adjustments. Accepted by JCT-
Regularly spaced subsums of integer partitions
For integer partitions , where , we study the sum of the parts of odd index. We show
that the average of this sum, over all partitions of , is of the
form More
generally, we study the sum of the parts whose
indices lie in a given arithmetic progression and we show that the average of
this sum, over all partitions of , is of the form
, with explicitly given
constants . Interestingly, for odd and we have
, so in this case the error term is of lower order. The methods used
involve asymptotic formulas for the behavior of Lambert series and the Zeta
function of Hurwitz.
We also show that if is the number of partitions of the sum of
whose parts of even index is , then for every , agrees with a
certain universal sequence, Sloane's sequence \texttt{#A000712}, for
but not for any larger
The motif problem
Fix a choice and ordering of four pairwise non-adjacent vertices of a
parallelepiped, and call a motif a sequence of four points in R^3 that coincide
with these vertices for some, possibly degenerate, parallelepiped whose edges
are parallel to the axes. We show that a set of r points can contain at most
r^2 motifs. Generalizing the notion of motif to a sequence of L points in R^p,
we show that the maximum number of motifs that can occur in a point set of a
given size is related to a linear programming problem arising from hypergraph
theory, and discuss some related questions.Comment: 17 pages, 1 figur
A Discontinuity in the Distribution of Fixed Point Sums
The quantity , defined as the number of permutations of the set
whose fixed points sum to , shows a sharp discontinuity
in the neighborhood of . We explain this discontinuity and study the
possible existence of other discontinuities in for permutations. We
generalize our results to other families of structures that exhibit the same
kind of discontinuities, by studying when ``fixed points'' is replaced
by ``components of size 1'' in a suitable graph of the structure. Among the
objects considered are permutations, all functions and set partitions.Comment: 1 figur
Asymptotic enumeration of correlation-immune boolean functions
A boolean function of boolean variables is {correlation-immune} of order
if the function value is uncorrelated with the values of any of the
arguments. Such functions are of considerable interest due to their
cryptographic properties, and are also related to the orthogonal arrays of
statistics and the balanced hypercube colourings of combinatorics. The {weight}
of a boolean function is the number of argument values that produce a function
value of 1. If this is exactly half the argument values, that is,
values, a correlation-immune function is called {resilient}.
An asymptotic estimate of the number of -variable
correlation-immune boolean functions of order was obtained in 1992 by
Denisov for constant . Denisov repudiated that estimate in 2000, but we will
show that the repudiation was a mistake.
The main contribution of this paper is an asymptotic estimate of
which holds if increases with within generous limits and specialises to
functions with a given weight, including the resilient functions. In the case
of , our estimates are valid for all weights.Comment: 18 page
Locally Restricted Compositions IV. Nearly Free Large Parts and Gap-Freeness
We define the notion of asymptotically free for locally restricted
compositions, which means roughly that large parts can often be replaced by any
larger parts. Two well-known examples are Carlitz and alternating compositions.
We show that large parts have asymptotically geometric distributions. This
leads to asymptotically independent Poisson variables for numbers of various
large parts. Based on this we obtain asymptotic formulas for the probability of
being gap free and for the expected values of the largest part, number of
distinct parts and number of parts of multiplicity k, all accurate to o(1).Comment: 28 page
Random Feedback Shift Registers, and the Limit Distribution for Largest Cycle Lengths
For a random binary noncoalescing feedback shift register of width , with
all possible feedback functions equally likely, the process
of long cycle lengths, scaled by dividing by , converges in distribution
to the same Poisson-Dirichlet limit as holds for random permutations in
, with all possible permutations equally likely. Such
behavior was conjectured by Golomb, Welch, and Goldstein in 1959.Comment: 42 pages; 21 references, 7 sections. The cover date, September 5,
2021, is hard-wired. Fixed some typos; added one more reference, and a new
Section 2, giving a survey of the proof of the main theore
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