The Combinatorics Workbench

This workbench collects interactive tools for the five branches of combinatorics: enumeration, graph theory, Ramsey theory, design theory, and coding theory. Each card targets a single technique, identity, or construction, with a widget that lets us vary the parameters and watch the resulting structure respond. Most cards include a note on a neighboring concept with which the technique is commonly conflated.

Factorial & Binomial

n r n! = 40320 · C(n,r) = 56

Pigeonhole Check

n m ⌈n/m⌉ = 4

Handshaking Check

degrees Σ=12, even ✓

Enumeration

counting techniques · 4 tools

Enumeration is the oldest and largest branch of combinatorics. The two foundations are the product rule and the sum rule; more sophisticated counts add bijections, generating functions, inclusion–exclusion, and the pigeonhole principle. Every count below decomposes into those foundations one way or another.

Binomial ↔ Multiset

ch03, ch05

Switch between choose r from n without repetition and choose r from n with repetition on the same row of balls; see the formulas transform.

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Pascal’s Triangle Heatmap

ch03

The first fifteen rows of Pascal’s triangle, shaded by magnitude. Click any cell to highlight its two parents and read off the defining identity.

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Inclusion–Exclusion Venn

ch10 §10.2

Two or three sets with adjustable sizes and intersections; the running inclusion–exclusion sum updates as the regions shift.

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Pigeonhole Demonstrator

ch10 §10.1

Place n pigeons into k holes; the fullest hole is guaranteed to hold at least ⌈n/k⌉, and the widget always exhibits a distribution realising that bound.

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Graph Theory

vertices & edges · 3 tools

A graph is a set of vertices with edges joining some pairs of vertices. The workhorse questions of graph theory are connectivity, traversal (Euler tours and Hamilton cycles), coloring, and planarity. Small examples carry most of the intuition, which is why the tools below operate on graphs of at most eight vertices.

Graph Sandbox

ch11–13

A fixed ring of vertices; click pairs to toggle edges. Live readouts cover degree sequence, the handshaking check, and connectedness.

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Euler vs. Hamilton

ch13

Two small graphs side by side, one tracing an Euler tour (edges visited once) and one tracing a Hamilton cycle (vertices visited once). The contrast is the point.

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Greedy Coloring

ch14 §14.3

Choose a vertex order and the widget greedy-colors the sandbox graph, reporting the upper bound χ ≤ Δ + 1 alongside the colors used.

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Ramsey Theory

inevitability of structure · 1 tool

Ramsey theory asserts that sufficiently large structures cannot avoid order. Concretely: any two-coloring of the edges of a sufficiently large complete graph contains a monochromatic clique of any chosen size. The threshold where this becomes unavoidable is the Ramsey number R(s, t), and most nontrivial values of R(s, t) remain unknown.

R(3,3) = 6

ch14 §14.2

Color each of the fifteen edges of K₆ red or blue by hand; the widget watches every triangle and flags the first monochromatic one that appears.

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Design Theory

structures of uniform regularity · 2 tools

A combinatorial design is a family of subsets of a base set, chosen so that small substructures appear with uniform frequency. Latin squares, Steiner triple systems, balanced incomplete block designs (BIBDs), and projective planes are the standard examples; they sit at the crossroads of counting, algebra, and finite geometry.

BIBD Parameters

ch17

Enter v, k, and λ; the widget derives r and b from the identities bk = vr and λ(v−1) = r(k−1), then reports whether Fisher’s inequality b ≥ v holds.

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Fano Plane

ch18

The seven points and seven blocks of the smallest Steiner triple system, drawn as a geometric diagram. Hovering a block highlights its three collinear points.

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Coding Theory

error correction from designs · 1 tool

An error-correcting code adds redundancy to a message so that a noisy channel cannot corrupt it past recovery. The book approaches codes through the combinatorial structures used to build them, with the Hamming code standing out as the canonical example of a code constructed from a projective plane.

Hamming (7,4) Code

ch19

Enter a four-bit message and the widget produces the seven-bit codeword; flip any single bit and the syndrome-decoding step recovers the original.

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