Combinatorics of Permutations, Second Edition

Permutations can be counted as linear orders, decomposed into cycles, tested for forbidden patterns, and studied as random objects. In Combinatorics of Permutations, Second Edition, Miklós Bóna follows these perspectives across a substantial treatment of modern combinatorics, connecting core counting ideas with algebra, probability, algorithms, and applications.

Several ways to see a permutation

The book begins with descents, Eulerian numbers, alternating runs, and subsequences, then turns to inversions and the major index. Its early chapters show how different ways of describing an ordering lead to distinct counting questions and useful generating functions.

From cycle structure to pattern avoidance

Readers next encounter cycle decompositions, cycle types, Stirling numbers, and tools such as the exponential formula and cycle index. A later sequence of chapters examines pattern avoidance—from basic patterns to broader results—and considers what happens when a pattern occurs repeatedly or a prescribed number of times.

Probability and algebraic connections

The treatment also takes a probabilistic view of permutations, including expectation, variance, standard deviation, and longest increasing subsequences. The algebraic perspective brings in the Robinson–Schensted–Knuth correspondence, permutation posets, and simplicial complexes.

For readers ready to work with the mathematics

This is a technical mathematics text for readers with a foundation appropriate to advanced combinatorics. Its wide-ranging topics make it relevant to students building depth in the subject and to researchers seeking connections among permutation enumeration, probability, algebra, and algorithms.

For a focused study of permutations and the structures they reveal, Bóna’s second edition offers a broad, carefully organized route through the field.