MahonianNumber

The number of permutations of $\{1, …, n\}$ with exactly $k$ inversions — entry $k$ of row $n$ of the Mahonian triangle.

MahonianNumber(n, k)the coefficient of

q^k

in the

q

-factorial

[n]_q!

CombinatoricaNumberOfPermutationsByInversions[n, k]

Typed signatures

Type	Package	Applies
`(integer, integer) -> integer`	`combinatorics`	always

Domain: Combinatorics

Details

Equals $Count(KInversionPermutations(n, k))$ , and the number of permutations $p$ with $Inversions(p) = k$ .
Satisfies $M(n, k) = \sum_{d < n} M(n - 1, k - d)$ and the symmetry $M(n, k) = M(n, \binom{n}{2} - k)$ . It is $0$ for $k > \binom{n}{2}$ .
Defined in Epsil as that recurrence over a table of rows; a bigint kernel answers, exactly, and tests/mahonian.test.ts holds the two together.
Combinatorica's NumberOfPermutationsByInversions; the same convention, with $k$ counted from $0$ . A negative $n$ or $k$ is left unevaluated, as compute-engine's Eulerian does (Combinatorica leaves a negative $k$ unevaluated too). The kernel also declines, rather than block, past about three million cell updates (about 0.2 s).
Not a core Wolfram head: there is no MahonianNumber in Wolfram 15 or the Function Repository.

Plot

2 more kept as data, checked against the oracles

Basic #

row $4$ : $1, 3, 5, 6, 5, 3, 1$

\mathrm{Map}(k↦\mathrm{MahonianNumber}(4, k), 0..1..6)

Out

\bigl\lbrack1, 3, 5, 6, 5, 3, 1\bigr\rbrack

Known value · OEIS A008302

the $15$ permutations of $5$ with $3$ inversions

\mathrm{MahonianNumber}(5, 3)

Out

15

Known value · wolframscript 15.0.0, Combinatorica NumberOfPermutationsByInversions[n, k]

the empty permutation has no inversions

\mathrm{MahonianNumber}(0, 0)

Out

1

Known value · wolframscript 15.0.0, Combinatorica NumberOfPermutationsByInversions[n, k]

$7 > \binom{3}{2}$

\mathrm{MahonianNumber}(3, 7)

Out

0

Known value · wolframscript 15.0.0, Combinatorica NumberOfPermutationsByInversions[n, k]

$M(6, 7) = 101$ both ways

\mathrm{MahonianNumber}(6, 7)=\mathrm{Count}(\mathrm{KInversionPermutations}(6, 7))

Out

\mathrm{True}

About

Examples	7
Checked against	Combinatorica 6 agree
Crosswalk	Wikipedia ×2 MathWorld Wikidata Encyclopedia of Mathematics nLab Britannica OEISCombinatorica

mappedcombinatoricapackages/oracle/src/mappings.ts

Combinatorica calls the Mahonian numbers NumberOfPermutationsByInversions.