Exercise 2092Combinatorics › Binomial coefficientsM1Added 3 Nov 2026

Mahonian numbers

#Mahonian#inversion#binomial

Statement

State the relationship between the Mahonian numbers (number of permutations with $k$ inversions) and the Gaussian binomial coefficients.

Hint

The Gaussian binomial $\binom{n}{k}_q$ counts $k$ -subsets of an $n$ -set, weighted by $q^{\text{sum of elements}}$ .

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Solution

Gaussian binomial. $\binom{n}{k}_q=\frac{(q^n-1)\cdots(q^{n-k+1}-1)}{(q^k-1)\cdots(q-1)}$ .

Mahonian numbers. The number of permutations of $n$ with $k$ inversions equals the coefficient of $q^k$ in $[n]_q!=\prod_{i=1}^n\frac{q^i-1}{q-1}$ .

Connection. The Gaussian binomial $\binom{n}{k}_q$ is a refinement of the binomial coefficient, with $q$ -analogue identities.

Conclusion. Mahonian numbers are the coefficients of the $q$ -factorial, connected to Gaussian binomials via $q$ -analogues of binomial identities.

Mahonian numbers

Statement

Solution

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