Math Stackexchange Posts
My home page is
here.
My home page at MSE is
here.
The OEIS played an essential part in
many of these computations.
Mellin transform computations
 A sum
formula by Hardy and Ramanujan
 A sum
formula by Cauchy and Ramanujan
 A
transform with remarkable symmetries
 Mellin Transform
Computation
 Working
with divergent Mellin Transforms

Asymptotics of $\sum_{k=1}^n \sqrt[4]{k}$

Asymptotics of $\sum_{k=1}^{n1} k^a$
 Plouffe,
Apery and Mellin
 A
functional equation relating two harmonic sums

Functional equation of the Hurwitz Zeta function

Two contrasting examples of fixed point scenarios
 Another
divergent Mellin transform
 Making
effective use of the fixed points of a functional equation to
evaluate a sum by Ramanujan

Double Bernoulli number
The collection is available here:
Not using the functional equation:
 A
Perron type formula
Polya and Burnside enumeration

Burnside lemma for orbital chromatic polynomials

Burnside lemma for K_n with an edge removed

Edge colorings of the ndimensional cube

Enumerating binary matrices with the symmetric group acting on
the rows and the cyclic group on the columns

Enumerating binary matrices with the symmetric group acting on
the rows and the symmetric group on the columns

Equivalence classes of matrices under row and column
permutation

Multisets of multisets from a given range with a repeated
element

Proper colorings of necklaces under rotational symmetry
(adjacent slots receive different colors)

Proper colorings of bracelets under rotational/reflectional
symmetry
(adjacent slots receive different colors)

Coloring bracelets properly, generating functions

Enumerating nonisomorphic graphs

Enumerating nonisomorphic
graphs, optimized

Enumerating nonisomorphic graphs with selfloops

Counting twocolorings of full rooted unordered binary trees
(child nodes may be swapped)

Burnside and PET when all elements of the repertoire must be
present at least once (Stirling numbers)

Polya enumeration, multplicative partitions of products of primes
and Stirling numbers of the second kind

Using the exponential formula to factor a simple
polynomial

Necklaces with the forbidden pattern 110

Primitive necklaces, the general numbertheoretic
formula

Bracelet with two pairs of N colors

A kind of set cover

A kind of set cover II

Strict partitions of fixed size with the set operator

Strict partitions of fixed size with the set operator, II

Coloring a board with toroidal symmetry

Fact sheet for necklaces and
bracelets

Coloring edges and vertices of necklaces and
bracelets simultaneously

Nonisomorphic bipartite graphs

Full symmetry of the cube (faces)
 Binary
necklaces with minimum maximum run length for black
 Binary
n X n X n tensor under rotational symmetry (cube)

Simultaneously coloring faces, vertices and edges of the cube
There is a list of Polya / Burnside topics at
MSE Meta.

Probability that a subset of k elements of [n] sums to a
multiple of n

Probability that a subset of n elements of [kn] sums to a
multiple of n

Probability that a multiset of k elements drawn from [n] sums to a
multiple of n

Generic algorithm for counting subsets of n elements of
[q] whose sum is divisible by some k

Symmetric polynomials

Unordered trees by leaves with outdegree at least two

Inequivalent colorings of a 2xN board under permutations of rows
and columns
The collection is available here:
Graphical enumeration

Enumerating labeled connected graphs

Enumerating labeled connected graphs by components

Enumerating labeled graphs with no endpoints

Enumerating unlabeled connected graphs
Cycle indices evaluated at harmonic numbers

Cycle index of the set operator, harmonic numbers and Stirling
numbers of the first kind

Cycle index of the multiset operator, harmonic numbers and complete
exponential Bell polynomials
Complex integration / contour integration
 Branches
of the logarithm

Recursively computing logarithmic integrals

Recursively computing logarithmic integrals, part two, a closed form

Replacing a branch of the logarithm in a CAS

Exercising complex algebra to compute a polyvalent trigonometric
integral

Exploiting the symmetry of a set of double poles to simplify
computation of the residue

Exploiting the symmetry of a set of double poles to simplify
computation of the residue II, somewhat more
computationintensive

Consistency in employing the chosen branch of the logarithm

Another complex integration applied to a trigonometric integral

Two branches of the logarithm

Two branches of the logarithm, second example

Two branches of the logarithm, third example

Two branches of the logarithm, fourth example

Two branches of the logarithm, computing an expansion at
infinity

A trigonometric sum

A parameterized integral cincluding a logarithm

A mixed integral including a square root

A mixed integral including a square root and a logarithm

A mixed integral including a trigonometric and a rational
term
Complex integration / cotangent multiplier and infinite
series
 The
trick with the $\pi\cot(\pi z)$ multiplier
 The
trick with the $\pi\cot(\pi z)$ multiplier II
 The
trick with the $\pi\cot(\pi z)$ multiplier III
Complex trigonometric integrals / contour integration
 A
trigonometric integral with three parameters
 Another
trigonometric integral with three parameters
 A
trigonometric integral with two parameters

Deformation of a standard trigonometric integral

A trigonometric integral with high order poles

Simple form of a betafunction type integral
Master theorem computations
 Bold challenge to the Master
Theorem
 Even bolder challenge to the Master
Theorem
 More Master Theorem
computations

Another Master Theorem computation

Master Theorem computation that produces a logarithmic factor

Master Theorem computation with Fibonacci numbers and the golden
ratio

A recurrence in two terms with considerable simplification
Faulhaber's formula
 Faulhaber's
formula
 Alternate
formulation of Faulhaber's formula in terms of Stirling numbers
 An
identity relating to Faulhaber's formula
 A
generalized form of Fauhlhaber's formula
 Another
identity relating to Faulhaber's formula
 Another
identity relating to Faulhaber's formula (falling
factorials)
Random permutations

Permutations of order two

Permutations of order k
 Fixed
points in permutations raised to a power
k

Permutations of odd order
 Trivariate
permutation generating function

Permutation generating function by total number of cycles and
fixed points
Stirling numbers
 Stirling
numbers, Eulerian numbers and a weighted power sum
 Stirling
numbers and Lah numbers, combinatorial classes
 Double
Stirling number convolution
 Barnes
integral, Stirling numbers and coefficient
asymptotics

Computing upper Stirling numbers in terms of Ward numbers

Double Stirling Number Identity
 Annihilated
coefficient extractors (ACE) and Stirling numbers
 Exponential
Generating Functions for Stirling numbers

Complex integration formula for converting an OGF into an EGF
 Stirling
number convolution with Bernoulli numbers
 Closed
form in terms of Stirling numbers of a binomial coefficient /
integer power convolution
 Closed
form in terms of Stirling numbers of a binomial coefficient /
integer power convolution II

Convolution of the two types of Stirling numbers

Stirling numbers of the first kind and the OGF of the cycle index
of the symmetric group

Stirling numbers of the first kind and permutations with two
fixed points and five cycles total

Stirling numbers of the second kind and a partial fraction
decomposition.

Stirling numbers of the second kind and a binomial sum

Stirling numbers of the second kind and a hypergeometric sum
 Expected
number of empty boxes in a balls into boxes problem (Stirling
numbers.).
 Expected
number of balls having at least one comrade in a balls into boxes
problem. (Modified Stirling numbers.)

Saddle point method applied to a Stirling number sum.

An intriguing Stirling number sum.
Coupon collector by Stirling numbers

Stirling numbers of the second kind and the coupon collector
problem (ACE technique)

Stirling numbers of the second kind and the coupon collector
problem (ACE technique), part two

Stirling numbers of the second kind and the coupon collector
problem (ACE technique), part three, expected number of
singletons

Investigating a coupon collector statistic (TchooseQ with T
number of steps and Q the different coupons present in the first j
coupons)

Drawing coupons in packets of q unique coupons

Drawing coupons until at least j instances of each type are
seen

Drawing coupons until at least 2 instances of each type are
seen, with n' types already collected

Drawing coupons until one instances of each type is
seen, with n' types already collected

Expected time until the first k coupons where k <= n have been
collected

Drawing coupons of n types with j instances of each type without
replacement

Drawing coupons of n types with j instances of each type without
replacement, fixed number of draws, number of types seen

Drawing coupons of n types with j instances of each type until all
of one type is seen, without replacement

Drawing coupons of n types with j instances of each type until two
of one type is seen, without replacement

Sum of coupon values when drawing coupons of n+1 types j with
nchoosej instances of each type without replacement
 Number
of tuples of some size present on completion with
replacement
The collection is available here:
Closely related:

Sampling D values without replacement once from G groups of X alike
values and seeing N different values
Balls and bins with exponential generating functions

Balls and bins with variable number of bins being drawn

Balls and bins where a specific configuration occurs

One bin of size k

Expected number of bins of size larger than k.

No singletons in bins.

Conditional expectations of number of singletons.
Eulerian numbers
 Double
Annihilated coefficient extractor (ACE) and Eulerian numbers
 Tricky
Eulerian number  polylog identity

Identity relating Catalan numbers, Bernoulli numbers, Eulerian
numbers and a counterpart of the Leibniz harmonic triangle
Principle of InclusionExclusion (PIE)

Matrices that do not share rows or columns with a template
matrix

Generating functions as a refinement of cardinalities, applied
to subsets of a set appearing

Permutations with no two consecutive entries (successor)

Circular permutations of multisets with no consecutive blocks
containing all instances of one type

A very simple ballsandbins probability

Drawing a sample from a multiset of colors without replacement,
probability all colors are present

Stirling numbers with no two consecutive values in the same subset

Permutations of colored balls with restrictions

Permutations of [n] with k mcycles
Lagrange inversion by residues
 Tree
enumeration by Lagrange Inversion A
 Tree
enumeration by Lagrange Inversion B
 Tree
enumeration by Lagrange Inversion C, two variables (rooted
labeled trees on n nodes by the number of leaves)
 Tree
enumeration by Lagrange Inversion CC, two variables (rooted
ordered trees on n nodes by the number of leaves)
 Tree
enumeration by Lagrange Inversion D, two variables
 A
Dary and Dregular trees
Rolling dice and exponential generating functions

Expected value of the number of appearances of the face that
ocurred most frequently

Average ratio of most frequent to least frequent

Number of distinct outcomes

Roll dice, sort, ask about the probability of a value appearing
at a given position

Number of faces that occured once

Rolling a die some number of times and observing a certain number of
values occur a certain number of times

Rolling a die while the values are nondecreasing, points are sum of
all values seen
Rolling dice and ordinary generating functions

Rolling a die some number of times and observing the number of
alternations between odd and even values

Rolling a die some number of times until a value has appeared k
times in a row
Qbinomial identities

Two qbinomial identities
Combinatorial sums and the Egorychev method
This has its own page.
These two links point to a curious
These links are Egorychevtype computations which are not by me /
involved significant external input:
Closely related
Using combstruct to enumerate trees
 A
parity bias for trees
 Number
of nodes with even offspring
 Trees
with odd degree sequence
 Average
depth of a leaf in an ordered binary tree
 Ordered
trees classified by the number of leaves, Catalan numbers, I
 Ordered
trees classified by the number of leaves, Catalan numbers, II
 A
class of ordered binary trees classified by the number of
leaves, Catalan numbers

Expected degree of the root in Cayley and Catalan trees

Internal path length in Cayley trees
Does not use combstruct directly but is closely related:
 Random
Mapping Statistics (cycle size and tail length)
 Random
Mapping Statistics (Pusieux series, acyclic graphs and connected
components)
 Unlabeled
trees branching into threenode leaves or sequences of at least
two subtrees
 Random
Mapping Statistics for f(f(x)) = f(f(f(x)))
 Number
of labelled trees on n vertices containing a fixed
edge
 Counting
labelled trees on n vertices by the number of leaves
 Number
of labelled trees on n vertices containing 2 fixed nonadjacent
edges

Asymptotics of a sum using singulariy analysis on rational function
of the tree function
 Labeled
unrooted trees with node degree one or three
 Labeled
unrooted trees with odd node degree
 Labeled
unrooted trees with odd node degree, simplified version
 Labeled
unrooted trees with unique vertex degree except for leaves
 Average
depth of a leaf in a binary tree
 Unlabeled
directed acyclic graphs with indegree at most one by PET

Enumerating labeled and unlabeled trees of some height h, species and
recurrence
 Two
recurrences for the number of unlabeled trees
 Unicyclic
connected graphs, labeled and unlabeled
 Trees
of maximum degree three, with three vertices of degree three,
labeled and unlabeled (NAUTY, Eiffel gadget)
 2regular
graphs, labeled
Analytic combinatorics

Probability that no couple sits together in a circle
(inclusionexclusion, Laplace's method)

Asymptotics of tree statistics for random mappings using Pusieux
expansion

Asymptotics of tree statistics for random mappings using Pusieux
expansion, tail length plus cycle size
Power Group Enumeration
 Power
Group Enumeration with two instances of the symmetric group by
Burnside
(canonical example)

Counting colored bicliques
 Power
Group Enumeration with to count necklaces with swappable colors
Burnside (canonical example)

Burnside lemma on nonisomorphic binary structures
(canonical example)
 Power
Group Enumeration of boolean functions under permutation and / or
complementation of inputs and complementation of outputs
 Power
Group Enumeration and Bell numbers
 Enumerating
hexagonal tiles by Power Group Enumeration and
Burnside
 Counting
binary structures

Primitive necklaces, the Mobius function, and Power Group Enumeration
 Edge
colorings of the cube and Power Group Enumeration, first
version
 Edge
colorings of the cube and Power Group Enumeration, optimized
version
 Face
colorings of the cube and Power Group Enumeration
 Counting
functions by Simultaneous Power Group Enumeration
 Subsets
of a standard 52 card deck wrt. suit permutation
(canonical example)

Stirling numbers and Power Group Enumeration

Binary matrices with a fixed number of ones per column under row
and column permutations

Sets of lottery tickets wrt. permutations of the values by the
symmetric group

Multisets of n permutations of [n] with the symmetric group acting on
the permutation elements

Colorings with interchangeable colors of an N by N square under
rotations and reflections

Coloring an N by M square wrt row and column permutations by the
symmetric group with the column permutations also acting on the
colors
 Set
partitions of unique elements from an nbym matrix where elements
from the same row may not be in the same partition

Number of sets of sequences of total length n over an alphabet of
n letters with the symmetric group acting on the letters.
 Stirling
numbers, the partition function, and multisets

Coloring bracelets with swappable colors, generating functions

Partitions of the multiset [1,1,2,2,...,n,n] into k submultisets

Bracelets containing k instances each of n swappable colors

Coloring the complete graph with swappable colors under graph
isomorphism
Coin flips

Probability of seeing t consecutive tails before h
consecutive heads from a fair coin (generating functions).

Expected number of coin flips of a biased coin until n
consecutive equal outcomes appear (generating functions).

Expected number of parallel coin flips of n biased coins until
every coin shows heads at least once.
generatingfunctionology by H. Wilf

Solving problem 1.20 on binary strings with restrictions on run
length.

Solving problem 2.25 on counting a type of colored
codeword.

Asymptotics of central Delannoy numbers.

Counting 2regular labeled graphs.

Asymptotics of Schroeder numbers / super Catalan numbers.
Peter Luschny's math
 The
PTransform produces Lah numbers and Stirling set / cycle numbers.
Miscellaneous

A Bell number identity

Multiple residues in a combinatorial sum

Multiple residues in a combinatorial sum (II)

Generalizing Catalan numbers

Permutations that don't contain adjacent (q,q+1)

Recurrence for number of partitions into distinct parts

Runs of bounded length in the rolls of a generic die

A Bernoulli number identity proved with complex variables

Generalizing Stirling numbers

Divergent series magic.

Asymptotics of a certain type of Smirnov words

An argument on a restricted Euler totient from basic number
theory.

Aces first set after drawing k cards from a standard
deck.

Generalization of a wellknown combinatorial identity.

Subsets not containing three consecutive elements (rational
generating functions).
 A Perl
script to generate Huffman codes.
 A
tricky digital sum in base 10

Expected maximum run length (consecutive elements) in a random subset
of m elements chosen from a total of n.

Rouche's theorem and Newton's method applied to forbidden
subsequences.

Translating a complicated regular expression for a language of ternary
strings into a rational generating function.

Expected number of runs of heads of length exactly k in a binary
string of length n.

Expected number of women next to at least one man in a seating
arrangement of n men and n women.

A tough combinatorial identity

A curious digital sum

First seeing k equal consecutive outcomes when sampling from m
equally likely ones

Justifying the multiplication of two binomials with noninteger
exponents using a formal power series argument

Two rounds of distributing n balls into 2n boxes with at most one ball
per box during a round

A somewhat unusual generating function from a Putnam problem

An elementary problem from number theory

Variance in a Bernoulli scheme

Partitions into distinct parts with no consecutive values

Sets of k distinct positive integers that sum to at most n

Complex number arithmetic with exponentials

A Stirling number inequality

NewtonGirard formulas

Contiguous runs of balls in a circular arrangement of bins

Partially balanced strings of parentheses and Catalan numbers