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

  1. A sum formula by Hardy and Ramanujan
  2. A sum formula by Cauchy and Ramanujan
  3. A transform with remarkable symmetries
  4. Mellin Transform Computation
  5. Working with divergent Mellin Transforms
  6. Asymptotics of $\sum_{k=1}^n \sqrt[4]{k}$
  7. Plouffe, Apery and Mellin
  8. A functional equation relating two harmonic sums
  9. Functional equation of the Hurwitz Zeta function
  10. Two contrasting examples of fixed point scenarios
  11. Another divergent Mellin transform
  12. Making effective use of the fixed points of a functional equation to evaluate a sum by Ramanujan
  13. Double Bernoulli number
The collection is available here: Not using the functional equation:
  1. A Perron type formula

Polya and Burnside enumeration

  1. Burnside lemma for orbital chromatic polynomials
  2. Burnside lemma for K_n with an edge removed
  3. Edge colorings of the n-dimensional cube
  4. Enumerating binary matrices with the symmetric group acting on the rows and the cyclic group on the columns
  5. Multisets of multisets from a given range with a repeated element
  6. Proper colorings of necklaces under rotational symmetry (adjacent slots receive different colors)
  7. Enumerating non-isomorphic graphs
  8. Enumerating non-isomorphic graphs, optimized
  9. Counting two-colorings of full rooted unordered binary trees (child nodes may be swapped)
  10. Burnside and PET when all elements of the repertoire must be present at least once (Stirling numbers)
  11. Polya enumeration, multplicative partitions of products of primes and Stirling numbers of the second kind
  12. Using the exponential formula to factor a simple polynomial
  13. Necklaces with the forbidden pattern 110
  14. Primitive necklaces, the general number-theoretic formula
  15. Bracelet with two pairs of N colors
  16. A kind of set cover
  17. A kind of set cover II
  18. Strict partitions of fixed size with the set operator
  19. Strict partitions of fixed size with the set operator, II
  20. Coloring a board with toroidal symmetry
  21. Fact sheet for necklaces and bracelets
  22. Non-isomorphic bipartite graphs
  23. Full symmetry of the cube (faces)
There is a list of Polya / Burnside topics at MSE Meta.

Exponential formula and subset / multiset sums

  1. Probability that a subset of k elements of [n] sums to a multiple of n
  2. Probability that a subset of n elements of [kn] sums to a multiple of n
  3. Probability that a multiset of k elements drawn from [n] sums to a multiple of n
  4. Generic algorithm for counting subsets of n elements of [q] whose sum is divisible by some k
The collection is available here:

Graphical enumeration

  1. Enumerating labeled connected graphs
  2. Enumerating labeled connected graphs by components
  3. Enumerating labeled graphs with no endpoints
  4. Enumerating unlabeled connected graphs

Cycle indices evaluated at harmonic numbers

  1. Cycle index of the set operator, harmonic numbers and Stirling numbers of the first kind
  2. Cycle index of the multiset operator, harmonic numbers and complete exponential Bell polynomials

Complex integration / contour integration

  1. Branches of the logarithm
  2. Recursively computing logarithmic integrals
  3. Recursively computing logarithmic integrals, part two, a closed form
  4. Replacing a branch of the logarithm in a CAS
  5. Exercising complex algebra to compute a polyvalent trigonometric integral
  6. Exploiting the symmetry of a set of double poles to simplify computation of the residue
  7. Exploiting the symmetry of a set of double poles to simplify computation of the residue II, somewhat more computation-intensive
  8. Consistency in employing the chosen branch of the logarithm
  9. Another complex integration applied to a trigonometric integral
  10. Two branches of the logarithm
  11. Two branches of the logarithm, second example
  12. Two branches of the logarithm, third example
  13. Two branches of the logarithm, computing an expansion at infinity
  14. A trigonometric sum
  15. A parameterized integral cincluding a logarithm
  16. A mixed integral including a square root
  17. A mixed integral including a square root and a logarithm

Complex integration / cotangent multiplier and infinite series

  1. The trick with the $\pi\cot(\pi z)$ multiplier
  2. The trick with the $\pi\cot(\pi z)$ multiplier II
  3. The trick with the $\pi\cot(\pi z)$ multiplier III

Complex trigonometric integrals / contour integration

  1. A trigonometric integral with three parameters
  2. Another trigonometric integral with three parameters
  3. A trigonometric integral with two parameters
  4. Deformation of a standard trigonometric integral
  5. A trigonometric integral with high order poles
  6. Simple form of a beta-function type integral

Master theorem computations

  1. Bold challenge to the Master Theorem
  2. Even bolder challenge to the Master Theorem
  3. More Master Theorem computations
  4. Another Master Theorem computation
  5. Master Theorem computation that produces a logarithmic factor
  6. Master Theorem computation with Fibonacci numbers and the golden ratio
  7. A recurrence in two terms with considerable simplification

Faulhaber's formula

  1. Faulhaber's formula
  2. Alternate formulation of Faulhaber's formula in terms of Stirling numbers
  3. An identity relating to Faulhaber's formula
  4. A generalized form of Fauhlhaber's formula
  5. Another identity relating to Faulhaber's formula

Random permutations

  1. Permutations of order k
  2. Fixed points in permutations raised to a power k
  3. Trivariate permutation generating function
  4. Permutation generating function by total number of cycles and fixed points

Stirling numbers

  1. Stirling numbers, Eulerian numberse and a weighted power sum
  2. Stirling numbers and Lah numbers, combinatorial classes
  3. Double Stirling number convolution
  4. Barnes integral, Stirling numbers and coefficient asymptotics
  5. Computing upper Stirling numbers in terms of Ward numbers
  6. Double Stirling Number Identity
  7. Annihilated coefficient extractors (ACE) and Stirling numbers
  8. Exponential Generating Functions for Stirling numbers
  9. Complex integration formula for converting an OGF into an EGF
  10. Stirling number convolution with Bernoulli numbers
  11. Closed form in terms of Stirling numbers of a binomial coefficient / integer power convolution
  12. Closed form in terms of Stirling numbers of a binomial coefficient / integer power convolution II
  13. Convolution of the two types of Stirling numbers
  14. Stirling numbers of the first kind and the OGF of the cycle index of the symmetric group
  15. Stirling numbers of the first kind and permutations with two fixed points and five cycles total
  16. Stirling numbers of the second kind and a partial fraction decomposition.
  17. Stirling numbers of the second kind and a binomial sum
  18. Stirling numbers of the second kind and a hypergeometric sum
  19. Expected number of empty boxes in a balls into boxes problem (Stirling numbers.).
  20. Expected number of balls having at least one comrade in a balls into boxes problem. (Modified Stirling numbers.)

Coupon collector by Stirling numbers

  1. Stirling numbers of the second kind and the coupon collector problem (ACE technique)
  2. Stirling numbers of the second kind and the coupon collector problem (ACE technique), part two
  3. Stirling numbers of the second kind and the coupon collector problem (ACE technique), part three, expected number of singletons
  4. Investigating a coupon collector statistic (T-choose-Q with T number of steps and Q the different coupons present in the first j coupons)
  5. Drawing coupons in packets of q unique coupons
  6. Drawing coupons until at least j instances of each type are seen
  7. Drawing coupons until at least 2 instances of each type are seen, with n' types already collected
  8. Drawing coupons until one instances of each type is seen, with n' types already collected
  9. Expected time until the first k coupons where k <= n have been collected
  10. Drawing coupons of n types with j instances of each type without replacement
  11. Drawing coupons of n types with j instances of each type without replacement, fixed number of draws, number of types seen
  12. Drawing coupons of n types with j instances of each type until all of one type is seen, without replacement
  13. Drawing coupons of n types with j instances of each type until two of one type is seen, without replacement
  14. Sum of coupon values when drawing coupons of n+1 types j with n-choose-j instances of each type without replacement
The collection is available here: Closely related:
  1. 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

  1. Balls and bins with variable number of bins being drawn
  2. Balls and bins where a specific configuration occurs
  3. One bin of size k
  4. Expected number of bins of size larger than k.

Eulerian numbers

  1. Double Annihilated coefficient extractor (ACE) and Eulerian numbers
  2. Tricky Eulerian number - polylog identity

Lagrange inversion formula

  1. Tree enumeration by Lagrange Inversion A
  2. Tree enumeration by Lagrange Inversion B
  3. Tree enumeration by Lagrange Inversion C, two variables (rooted labeled trees on n nodes by the number of leaves)
  4. Tree enumeration by Lagrange Inversion D, two variables
  5. A remarkably simple result from Lagrange Inversion
  6. A D-ary and D-regular trees

Inclusion-Exclusion

  1. Matrices that do not share rows or columns with a template matrix
  2. Generating functions as a refinement of cardinalities, applied to subsets of a set appearing
  3. Ternary words that avoid the pattern 22
  4. Permutations with no two consecutive entries (successor)
  5. Circular permutations of multisets with no consecutive blocks containing all instances of one type
  6. A very simple balls-and-bins probability
  7. Drawing a sample from a multiset of colors without replacement, probability all colors are present

Rolling dice and exponential generating functions

  1. Expected value of the number of appearances of the face that ocurred most frequently
  2. Average ratio of most frequent to least frequent
  3. Number of distinct outcomes
  4. Roll dice, sort, ask about the probability of a value appearing at a given position
  5. Number of faces that occured once
  6. Rolling a die some number of times and observing a certain number of values occur a certain number of times
  7. Rolling a die while the values are non-decreasing, points are sum of all values seen

Rolling dice and ordinary generating functions

  1. Rolling a die some number of times and observing the number of alternations between odd and even values
  2. Rolling a die some number of times until a value has appeared k times in a row

Binomial coefficients and Egorychev method

This has its own page. These two links point to a curious These links are Egorychev-type computations which are not by me / involved significant external input: Closely related

Using combstruct to enumerate trees

  1. A parity bias for trees
  2. Number of nodes with even offspring
  3. Trees with odd degree sequence
  4. Average depth of a leaf in an ordered binary tree
  5. Ordered trees classified by the number of leaves, Catalan numbers, I
  6. Ordered trees classified by the number of leaves, Catalan numbers, II
  7. A class of ordered binary trees classified by the number of leaves, Catalan numbers
Does not use combstruct directly but is closely related:
  1. Random Mapping Statistics (cycle size and tail length)
  2. Unlabeled trees branching into three-node leaves or sequences of at least two subtrees
  3. Random Mapping Statistics for f(f(x)) = f(f(f(x)))
  4. Number of labelled trees on n vertices containing a fixed edge
  5. Counting labelled trees on n vertices by the number of leaves
  6. Number of labelled trees on n vertices containing 2 fixed non-adjacent edges
  7. Asymptotics of a sum using singulariy analysis on rational function of the tree function
  8. Labeled unrooted trees with node degree one or three
  9. Labeled unrooted trees with odd node degree
  10. Labeled unrooted trees with odd node degree, simplified version
  11. Labeled unrooted trees with unique vertex degree except for leaves
  12. Average depth of a leaf in a binary tree
  13. Unlabeled directed acyclic graphs with indegree at most one by PET
  14. Enumerating labeled and unlabeled trees of some height h, species and recurrence
  15. Two recurrences for the number of unlabeled trees
  16. Unicyclic connected graphs, labeled and unlabeled

Analytical combinatorics

  1. Probability that no couple sits together in a circle (inclusion-exclusion, Laplace's method)

Power Group Enumeration

  1. Power Group Enumeration with two instances of the symmetric group by Burnside (canonical example)
  2. Counting colored bicliques
  3. Power Group Enumeration with to count necklaces with swappable colors Burnside (canonical example)
  4. Burnside lemma on non-isomorphic binary structures (canonical example)
  5. Power Group Enumeration of boolean functions under permutation and / or complementation of inputs and complementation of outputs
  6. Power Group Enumeration and Bell numbers
  7. Enumerating hexagonal tiles by Power Group Enumeration and Burnside
  8. Counting binary structures
  9. Primitive necklaces, the Mobius function, and Power Group Enumeration
  10. Edge colorings of the cube and Power Group Enumeration, first version
  11. Edge colorings of the cube and Power Group Enumeration, optimized version
  12. Counting functions by Simultaneous Power Group Enumeration
  13. Subsets of a standard 52 card deck wrt. suit permutation (canonical example)
  14. Stirling numbers and Power Group Enumeration
  15. Binary matrices with a fixed number of ones per column under row and column permutations
  16. Sets of lottery tickets wrt. permutations of the values by the symmetric group
  17. Multisets of n permutations of [n] with the symmetric group acting on the permutation elements
  18. Colorings with interchangeable colors of an N by N square under rotations and reflections
  19. Coloring an N by M square wrt row and column permutations by the symmetric group with the column permutations also acting on the colors
  20. Set partitions of unique elements from an n-by-m matrix where elements from the same row may not be in the same partition
  21. Number of sets of sequences of total length n over an alphabet of n letters with the symmetric group acting on the letters.
  22. Stirling numbers, the partition function, and multisets

Coin flips

  1. Probability of seeing t consecutive tails before h consecutive heads from a fair coin (generating functions).
  2. Expected number of coin flips of a biased coin until n consecutive equal outcomes appear (generating functions).
  3. Expected number of parallel coin flips of n biased coins until every coin shows heads at least once.

generatingfunctionology by H. Wilf

  1. Solving problem 1.20 on binary strings with restrictions on run length.
  2. Solving problem 2.25 on counting a type of colored codeword.

Miscellaneous

  1. A Bell number identity
  2. Multiple residues in a combinatorial sum
  3. Multiple residues in a combinatorial sum (II)
  4. Generalizing Catalan numbers
  5. Permutations that don't contain adjacent (q,q+1)
  6. Recurrence for number of partitions into distinct parts
  7. Runs of bounded length in the rolls of a generic die
  8. A Bernoulli number identity proved with complex variables
  9. Generalizing Stirling numbers
  10. Divergent series magic.
  11. Asymptotics of a certain type of Smirnov words
  12. An argument on a restricted Euler totient from basic number theory.
  13. Aces first set after drawing k cards from a standard deck.
  14. Generalization of a well-known combinatorial identity.
  15. Subsets not containing three consecutive elements (rational generating functions).
  16. A Perl script to generate Huffman codes.
  17. A tricky digital sum in base 10
  18. Expected maximum run length (consecutive elements) in a random subset of m elements chosen from a total of n.
  19. Rouche's theorem and Newton's method applied to forbidden subsequences.
  20. Translating a complicated regular expression for a language of ternary strings into a rational generating function.
  21. Expected number of runs of heads of length exactly k in a binary string of length n.
  22. Expected number of women next to at least one man in a seating arrangement of n men and n women.
  23. A tough combinatorial identity
  24. A curious digital sum
  25. First seeing k equal consecutive outcomes when sampling from m equally likely ones
  26. Justifying the multiplication of two binomials with noninteger exponents using a formal power series argument
  27. Two rounds of distributing n balls into 2n boxes with at most one ball per box during a round
  28. A somewhat unusual generating function from a Putnam problem
  29. An elementary problem from number theory
  30. Variance in a Bernoulli scheme