Enumerative Combinatorics
Skills
Every skill in the course, and how you are doing on each.
Counting principles, bijections, and the pigeonhole principle
- Drill
Overcounting division
- Drill
Product rule
- Drill
Complementary counting
- Drill
Sum rule
- Drill
Words with restrictions
- Drill
Circular arrangements
- Drill
Bijective counting
- Drill
Pigeonhole principle
- Drill
Combinatorial proof
- Drill
Double counting
- Drill
Distinguished representative
- Drill
Symmetry
- Drill
Explicit bijection
- Drill
Parity
72 more topics with one problem
- Tagged objects
- Fixed size selections
- Gap method
- Extremal counting
- Residue classes
- Rotational symmetry
- Disjoint teams
- Conditional choices
- Restricted words
- Equivalence classes
- Fixed points
- Involution
- Permutation descents
- Binary encoding
- Gray code
- Function fibres
- Left right inverses
- Elementwise encoding
- Nested subsets
- Graph degrees
- Distinct counts
- Recursive bijection
- Tournaments
- Sharp bound
- Spacing
- Chain decomposition
- Extremal subsets
- Forbidden ratios
- Consecutive blocks
- Prefix sums
- Residues
- Capacities
- Generalized pigeonhole
- Sharp threshold
- Binary words
- Marked occurrences
- Overlapping patterns
- Extremal construction
- Monotone subsequences
- Edge colourings
- Sharp example
- Fractional parts
- Rational approximation
- Distinct subsequences
- Duplicate subtraction
- Previous occurrence
- Euler tour
- Shortest cover
- Universal binary cycles
- Grid rectangles
- Side lengths
- Even subgraphs
- Forced completion
- Spanning tree
- Binary vectors
- Changing choice counts
- Independent columns
- Forced images
- Iterated functions
- Root fibre
- Cyclic minimum
- Plane tree profile
- Preorder slots
- Marked orders
- Tournament paths
- Vertex insertion
- Cycle orientations
- Cyclic transitions
- Source count
- Initial component
- Recurrence decomposition
- Strong tournaments
Binomial and multinomial coefficients, identities, and lattice paths
- Drill
Coefficient comparison
- Drill
Sign reversing involution
- Drill
Binomial inversion
- Drill
Polynomial identity
- Drill
Membership regions
72 more topics with one problem
- Complement bijection
- Factorial formula
- Subset counting
- Boundary conventions
- Pascal identity
- Recursive counting
- Hockey stick identity
- Largest element
- Two class selection
- Vandermonde identity
- Multinomial counting
- Repeated symbols
- Sequential selection
- Multinomial expansion
- Polynomial coefficients
- Signed weights
- Lattice paths
- Path concatenation
- Prescribed vertices
- Inverse map
- One sided barrier
- Reflection principle
- Prefix inequality
- Strict ballot
- Weak versus strict
- Determinant count
- Nonintersecting paths
- Tail switching
- Alternating sums
- Truncated sum
- Alternating cancellation
- Triangular transform
- Adjacent ratios
- Log concavity
- Unimodality
- Antichains
- Permutation chains
- Weighted double counting
- Binary digits
- Polynomial modulo two
- Alternating binomial sums
- Finite differences
- Polynomial cancellation
- Parity cancellation
- Signed convolution
- Balancing transfer
- Inventory vectors
- Multinomial maximum
- Roots of unity filter
- Subset size residues
- Weighted expansion
- Finite vandermonde
- Generalized binomial
- Adjacent row ratios
- Normalized recurrence
- Reciprocal binomial sum
- Ordered pairs
- Subset intersections
- Distinguished element
- Reciprocal weights
- Lattice path area
- Reversal pairing
- Total statistic
- Endpoint classification
- Exchangeable gaps
- Subset span
- Delannoy paths
- Diagonal step count
- Recurrence verification
- Basis quotient
- Binary subspaces
- Prescribed vector
Compositions, multisets, and distributing objects into boxes
- Drill
Positive compositions
68 more topics with one problem
- Empty boxes
- Stars and bars
- Weak compositions
- Cut positions
- Variable length
- Feasibility
- Lower bounds
- Variable shift
- Slack variable
- Total upper bound
- Gap removal
- Multiset selection
- Sorted encoding
- Distinguishability
- Labeled boxes
- Occupancy patterns
- Nonempty boxes
- Positive occupancies
- Support selection
- Congruence restrictions
- Variable substitution
- Bounded parts
- Capacity constraints
- Inclusion exclusion application
- Capacity complement
- Deficit counting
- Symmetric totals
- Paired coordinates
- Palindromic compositions
- Parity constraint
- Exact multiplicity
- Unit parts
- First part recurrence
- Fixed length count
- Odd parts
- Gap encoding
- Marked stars
- Weighted composition sum
- Endpoint types
- Lattice path turns
- Positive runs
- Binary runs
- Positive and weak parts
- Separator gaps
- Empty box symmetry
- Occupancy profile
- Unlabeled boxes
- Model selection
- Object and box labels
- Twelvefold way
- Common refinement
- Composition refinement
- Cut set structure
- Diagonal count
- Endpoint comparison
- Reversal symmetry
- Circular parking
- Parking functions
- Rotation bijection
- Circular compositions
- Fixed rotations
- Orbit averaging
- Adjacent unequal parts
- Finite recurrence
- Terminal state
- Labeled ball collisions
- Marked pairs
- Occupancy squares
Inclusion-exclusion, derangements, and surjections
72 more topics with one problem
- Overlapping intersections
- Three set counting
- Pointwise cancellation
- Sieve formula
- Weighted counting
- Exact membership sieve
- Intersection totals
- Alternating error
- Bonferroni bounds
- Truncated sieve
- Alternating series error
- Derangements
- Forbidden fixed points
- Derangement recurrence
- Two case bijection
- Disjoint classification
- Exact fixed points
- Rencontres count
- Compatible constraints
- Forbidden positions
- Rook numbers
- Disjoint blocks
- Rook polynomial
- Cycle matchings
- Cyclic forbidden positions
- Rook sieve
- Labeled functions
- Omitted images
- Surjections
- Optional symbols
- Partial surjectivity
- Required symbols
- Empty or singleton
- Labeled occupancy
- Minimum fibre size
- Euler totient
- Prime divisibility
- Sieve product
- Coprime pairs
- Mobius sieve
- Floor counts
- Least common multiple
- Truncated divisibility
- Connected components
- Edge sieve
- Proper colourings
- Block compression
- Disjoint pairs
- Forbidden adjacencies
- Chain components
- Directed adjacencies
- Overlapping blocks
- Binary matrices
- Intersecting zero constraints
- Nonzero rows and columns
- Forced edges
- Overlapping triangles
- Triangle free graphs
- Missing code labels
- Prescribed support
- Tree leaves
- Isolated vertices
- Labeled graphs
- Vertex sieve
- Divisor periods
- Prime sieve
- Primitive words
- Primitive compositions
- Sum constrained sieve
- Parity pairing
- Signed derangements
- Weighted sieve
Set partitions, Stirling numbers, and the cycle structure of permutations
- Drill
Permutation powers
69 more topics with one problem
- Boundary values
- Last element recurrence
- Stirling second kind
- Canonical block labels
- Restricted growth strings
- Block size profile
- Equal size blocks
- Factorial division
- Distinguished representatives
- Idempotent functions
- Pointed blocks
- Bell numbers
- Distinguished block
- Subset recurrence
- Falling factorial basis
- Fibre partitions
- Cycle notation
- Inverse permutation
- Last label recurrence
- Stirling first kind
- Cycle type counting
- Repeated cycle lengths
- Rotations within cycles
- Cycle count polynomial
- Rising factorial
- Signed evaluation
- Distinguished label
- Marked cycle
- Total cycle count
- Involutions
- Recurrence
- Singleton pair decomposition
- Allowed cycle lengths
- Fixed point count
- Polynomial bases
- Signed stirling inversion
- Triangular matrices
- Inversion vector
- Lehmer code
- Polynomial enumerator
- Adjacent swaps
- Cycle sign
- Inversion parity
- Descent recurrence
- Eulerian numbers
- Maximum insertion
- Increasing runs
- One descent
- Subset encoding
- Canonical form
- Cycle record bijection
- Left to right maxima
- Cyclic descents
- Eulerian distribution
- Rotation orbits
- Invariant partitions
- Label symmetry
- Cycle interleaving
- Permutation roots
- Block refinement
- Noncrossing partitions
- Maximal chains
- Partition lattice
- Commuting permutations
- Cycle automorphisms
- Bell congruences
- Prime cycle action
- Prescribed descents
- Subset inversion
Integer partitions and Ferrers diagrams
- Drill
Unique factorization
62 more topics with one problem
- Exhaustive listing
- Length distribution
- Partition definition
- Ferrers conjugation
- Row column duality
- Size preservation
- Boundary conditions
- Exact length recurrence
- Remove one
- Column removal
- Largest part bound
- Length bound
- Boundary path count
- Bounded rectangle
- Largest part recurrence
- Coefficient symmetry
- Rectangle complement
- Self complementary shapes
- Distinct parts
- Minimum size
- Staircase removal
- Binary multiplicity
- Odd distinct bijection
- Base r bijection
- Bounded multiplicity
- Forbidden multiples
- Diagonal hooks
- Distinct odd parts
- Self conjugate shapes
- Durfee square
- Independent remainders
- Size convolution
- Conjugate multiplicities
- Diagram scaling
- Divisible part sizes
- Allowed part sizes
- Euler product
- Formal coefficients
- Bivariate partition series
- Fixed length product
- Length marking
- Bounded multiplicity series
- Coefficient extraction
- Finite geometric factors
- Divisor sums
- Marked part occurrences
- Partition recurrence
- Gaussian binomial
- Rational product proof
- Rectangle polynomial
- Reflection bijection
- Young tableaux
- Conjugate diagrams
- Dominance order
- Marked parts
- Overpartitions
- Product bijection
- Pentagonal numbers
- Finite sums
- Plane partitions
- Multiset blocks
- Vector partitions
Ordinary generating functions, recurrences, and Catalan numbers
57 more topics with one problem
- Formal series
- Partial sums
- Shifted sequences
- Additive size
- Cauchy product
- Convolution count
- Colored tiles
- Sequence construction
- Series composition
- Characteristic roots
- Recurrence initial terms
- Simple poles
- Exact coefficients
- Inhomogeneous recurrence
- Polynomial forcing
- Coefficient formula
- Polynomial times exponential
- Repeated poles
- Eventual recurrence
- Rational series equivalence
- Repeated characteristic root
- Binet formula
- Fibonacci partial sums
- Fibonacci series
- Doubling formulas
- Fibonacci addition
- Recurrence uniqueness
- Cassini identity
- Consecutive coprimality
- Fibonacci matrix
- Binomial transform series
- Fibonacci even indices
- Formal composition
- Forbidden word automaton
- Rational word series
- Suffix states
- Catalan functional equation
- Dyck first return
- Formal square root
- Catalan coefficient extraction
- Central binomial coefficient
- Formal differentiation
- Leaf count
- Ordered full binary trees
- Tree dyck bijection
- Catalan recurrence
- Convex polygon triangulations
- Root edge decomposition
- Continued fraction recurrence
- Height bounded dyck words
- Rational catalan restriction
- Binary index pattern
- Catalan parity
- Formal series mod two
- Indecomposable permutations
- Formal residues
- Lagrange inversion
Exponential generating functions and labeled structures
45 more topics with one problem
- Factorial normalization
- Label allocation
- Labeled product
- Distinguished extra label
- Egf derivative
- Pointing
- Exponential formula
- Free block ordering
- Unordered components
- Bell bivariate series
- Block count marker
- Stirling egf
- Distinguished block recurrence
- Exponential restriction
- Singleton free partitions
- Factorial profile count
- Pairs and triples
- Restricted block series
- Named fibres
- Occupancy restrictions
- Surjection egf
- Alternating recurrence
- Derangement egf
- Forbidden singleton cycles
- Fixed point marker
- Involution bivariate egf
- Marked label average
- Cycle size restriction
- Differential recurrence
- Odd cycle permutations
- Connected labeled graphs
- Distinguished component
- Logarithm of assemblies
- Internal block order
- Lah numbers
- Unordered set of lists
- First block recurrence
- Ordered set partitions
- Sequence of blocks
- Average block count
- Factorial moments
- Marked components
- Egf differential equation
- Increasing binary trees
- Minimum root bijection