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OEIS Wiki

Table of contents

1. Mellin transform computations
2. Polya and Burnside enumeration
3. Exponential formula and set sums
4. Graphical enumeration
5. Cycle indices and harmonic numbers
6. Complex integration
7. Cotangent multiplier
8. Complex integration / trigonometry
9. Master theorem
10. Faulhaber's formula
11. Random permutations
12. Stirling numbers
13. Coupon collector by Stirling numbers
14. EGFs for balls and bins
15. Eulerian numbers
16. Principle of Inclusion-Exclusion
17. Lagrange Inversion by Residues
18. EGFs for rolling dice
19. OGFs for rolling dice
20. Q-binomial identities
21. Egorychev method
22. combstruct
23. Analytic Combinatorics
24. Power Group Enumeration
25. Coinflips
26. generatingfunctionology
27. Peter Luschny's math
28. Combinatorics on words
29. Miscellaneous

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. Asymptotics of $\sum_{k=1}^{n-1} k^a$
8. Plouffe, Apery and Mellin
9. A functional equation relating two harmonic sums
10. Functional equation of the Hurwitz Zeta function
11. Two contrasting examples of fixed point scenarios
12. Another divergent Mellin transform
13. Making effective use of the fixed points of a functional equation to evaluate a sum by Ramanujan
14. Double Bernoulli number

• Mellin transform and the functional equation of the Riemann Zeta function (PDF)
and here:
• Mellin transform and the functional equation of the Riemann Zeta function (PS.GZ)

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. Enumerating binary matrices with the symmetric group acting on the rows and the symmetric group on the columns
6. Equivalence classes of matrices under row and column permutation
7. Multisets of multisets from a given range with a repeated element
8. Proper colorings of necklaces under rotational symmetry (adjacent slots receive different colors)
9. Proper colorings of bracelets under rotational/reflectional symmetry (adjacent slots receive different colors)
10. Coloring bracelets properly, generating functions
11. Enumerating non-isomorphic graphs
12. Enumerating non-isomorphic graphs, optimized
13. Enumerating non-isomorphic graphs with self-loops
14. Counting two-colorings of full rooted unordered binary trees (child nodes may be swapped)
15. Burnside and PET when all elements of the repertoire must be present at least once (Stirling numbers)
16. Polya enumeration, multplicative partitions of products of primes and Stirling numbers of the second kind
17. Using the exponential formula to factor a simple polynomial
18. Necklaces with the forbidden pattern 110
19. Primitive necklaces, the general number-theoretic formula
20. Bracelet with two pairs of N colors
21. A kind of set cover
22. A kind of set cover II
23. Strict partitions of fixed size with the set operator
24. Strict partitions of fixed size with the set operator, II
25. Coloring a board with toroidal symmetry
26. Fact sheet for necklaces and bracelets
27. Coloring edges and vertices of necklaces and bracelets simultaneously
28. Non-isomorphic bipartite graphs
29. Full symmetry of the cube (faces)
30. Binary necklaces with minimum maximum run length for black
31. Binary n X n X n tensor under rotational symmetry (cube)
32. Simultaneously coloring faces, vertices and edges of the cube

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
5. Symmetric polynomials
6. Unordered trees by leaves with outdegree at least two
7. Inequivalent colorings of a 2xN board under permutations of rows and columns

• Cycle indices from the exponential formula and subset / multiset sums (PDF)
and here:
• Cycle indices from the exponential formula and subset / multiset sums (PS.GZ)

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, fourth example
14. Two branches of the logarithm, computing an expansion at infinity
15. A trigonometric sum
16. A parameterized integral cincluding a logarithm
17. A mixed integral including a square root
18. A mixed integral including a square root and a logarithm
19. A mixed integral including a trigonometric and a rational term

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
6. Another identity relating to Faulhaber's formula (falling factorials)

Random permutations

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

Stirling numbers

1. Stirling numbers, Eulerian numbers 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.)
21. Saddle point method applied to a Stirling number sum.
22. An intriguing Stirling number sum.

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
15. Number of tuples of some size present on completion with replacement

• Coupon collector and Stirling numbers (PDF)
and here:
• Coupon collector and Stirling numbers (PS.GZ)

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.
5. No singletons in bins.
6. Conditional expectations of number of singletons.

Eulerian numbers

1. Double Annihilated coefficient extractor (ACE) and Eulerian numbers
2. Tricky Eulerian number - polylog identity
3. Identity relating Catalan numbers, Bernoulli numbers, Eulerian numbers and a counterpart of the Leibniz harmonic triangle

Principle of Inclusion-Exclusion (PIE)

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. Permutations with no two consecutive entries (successor)
4. Circular permutations of multisets with no consecutive blocks containing all instances of one type
5. A very simple balls-and-bins probability
6. Drawing a sample from a multiset of colors without replacement, probability all colors are present
7. Stirling numbers with no two consecutive values in the same subset
8. Permutations of colored balls with restrictions
9. Permutations of [n] with k m-cycles
10. Labeled trees with k leaves
11. Numbers where a certain set of digits appears

Lagrange inversion by residues

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 CC, two variables (rooted ordered trees on n nodes by the number of leaves)
5. Tree enumeration by Lagrange Inversion D, two variables
6. A D-ary and D-regular trees
7. Inverting a series to produce C(an,n)/a/n

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

Q-binomial identities

1. Two q-binomial identities

Combinatorial sums and the Egorychev method

• Polya enumeration and a binomial coefficient convolution
• Polya enumeration and a binomial coefficient convolution, sequel

• Binomial coefficient convolution
• A triple hypergeometric
• A trigonometric sum
• An infinite series

• Partial fractions by residues and a multiple binomial coefficient sum
• A binomial coefficient / polynomoial convolution
• A very curious binomial coefficient identity.
• A very curious binomial coefficient identity, II.
• An interesting binomial coefficient identity.

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
8. Expected degree of the root in Cayley and Catalan trees
9. Internal path length in Cayley trees

1. Random Mapping Statistics (cycle size and tail length)
2. Random Mapping Statistics (Pusieux series, acyclic graphs and connected components)
3. Unlabeled trees branching into three-node leaves or sequences of at least two subtrees
4. Random Mapping Statistics for f(f(x)) = f(f(f(x)))
5. Number of labelled trees on n vertices containing a fixed edge
6. Counting labelled trees on n vertices by the number of leaves
7. Number of labelled trees on n vertices containing 2 fixed non-adjacent edges
8. Asymptotics of a sum using singulariy analysis on rational function of the tree function
9. Labeled unrooted trees with node degree one or three
10. Labeled unrooted trees with odd node degree
11. Labeled unrooted trees with odd node degree, simplified version
12. Labeled unrooted trees with unique vertex degree except for leaves
13. Average depth of a leaf in a binary tree
14. Unlabeled directed acyclic graphs with indegree at most one by PET
15. Enumerating labeled and unlabeled trees of some height h, species and recurrence
16. Two recurrences for the number of unlabeled trees
17. Unicyclic connected graphs, labeled and unlabeled
18. Trees of maximum degree three, with three vertices of degree three, labeled and unlabeled (NAUTY, Eiffel gadget)
19. 2-regular graphs, labeled

Analytic combinatorics

1. Probability that no couple sits together in a circle (inclusion-exclusion, Laplace's method)
2. Asymptotics of tree statistics for random mappings using Pusieux expansion
3. Asymptotics of tree statistics for random mappings using Pusieux expansion, tail length plus cycle size

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. Face colorings of the cube and Power Group Enumeration
13. Counting functions by Simultaneous Power Group Enumeration
14. Subsets of a standard 52 card deck wrt. suit permutation (canonical example)
15. Stirling numbers and Power Group Enumeration
16. Binary matrices with a fixed number of ones per column under row and column permutations
17. Sets of lottery tickets wrt. permutations of the values by the symmetric group
18. Multisets of n permutations of [n] with the symmetric group acting on the permutation elements
19. Colorings with interchangeable colors of an N by N square under rotations and reflections
20. Coloring an N by M square wrt row and column permutations by the symmetric group with the column permutations also acting on the colors
21. Set partitions of unique elements from an n-by-m matrix where elements from the same row may not be in the same partition
22. Number of sets of sequences of total length n over an alphabet of n letters with the symmetric group acting on the letters.
23. Stirling numbers, the partition function, and multisets
24. Coloring bracelets with swappable colors, generating functions
25. Partitions of the multiset [1,1,2,2,...,n,n] into k submultisets
26. Bracelets containing k instances each of n swappable colors
27. Coloring the complete graph with swappable colors under graph isomorphism

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.
3. Asymptotics of central Delannoy numbers.
4. Counting 2-regular labeled graphs.
5. Asymptotics of Schroeder numbers / super Catalan numbers.

Peter Luschny's math

1. The P-Transform produces Lah numbers and Stirling set / cycle numbers.

Combinatorics on words

1. Number of binary strings with an even number of adjacent ones
2. Words with at least one run of some length
3. Asymptotics of a certain type of Smirnov words
4. Runs of bounded length in the rolls of a generic die
5. Expected number of runs of heads of length exactly k in a binary string of length n.

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. A Bernoulli number identity proved with complex variables
8. Generalizing Stirling numbers
9. Divergent series magic.
10. An argument on a restricted Euler totient from basic number theory.
11. Aces first set after drawing k cards from a standard deck.
12. Generalization of a well-known combinatorial identity.
13. Subsets not containing three consecutive elements (rational generating functions).
14. A Perl script to generate Huffman codes.
15. A tricky digital sum in base 10
16. Expected maximum run length (consecutive elements) in a random subset of m elements chosen from a total of n.
17. Rouche's theorem and Newton's method applied to forbidden subsequences.
18. Translating a complicated regular expression for a language of ternary strings into a rational generating function.
19. Expected number of women next to at least one man in a seating arrangement of n men and n women.
20. A tough combinatorial identity
21. A curious digital sum
22. First seeing k equal consecutive outcomes when sampling from m equally likely ones
23. Justifying the multiplication of two binomials with noninteger exponents using a formal power series argument
24. Two rounds of distributing n balls into 2n boxes with at most one ball per box during a round
25. A somewhat unusual generating function from a Putnam problem
26. An elementary problem from number theory
27. Variance in a Bernoulli scheme
28. Partitions into distinct parts with no consecutive values
29. Sets of k distinct positive integers that sum to at most n
30. Complex number arithmetic with exponentials
31. A Stirling number inequality
32. Newton-Girard formulas
33. Contiguous runs of balls in a circular arrangement of bins
34. Partially balanced strings of parentheses and Catalan numbers
35. Counting zygodromes