Mathematica code for q-Stirling numbers

In the paper [A new $$q$$-Analog of Stirling Numbers],(https://hal.archives-ouvertes.fr/hal-01372920/document)[PDF] J. Cigler defined $$q$$-Stirling numbers of the second kind as the following:

He considered the weight $$w(\pi)$$ for each partition $$\pi$$ of the set $$A=\{0,\cdots,n-1\}$$, and distinguished the part containing the zero element and called it $$B_0$$. Then,

$$w(\pi)=q^{\sum_{i\in B_0} i}$$ For each set $$A$$ of partitions, let $$w(A)=\sum _{\pi\in A}w(\pi)$$. Also, he considered $$A_{n,k}$$ to be the set of all partitions of $$\{0,\cdots,n-1\}$$ into $$k$$ non-empty parts. So $${n\brace k}=w(A_{n,k})$$

My question is:

What Mathematica code would generate these Stirling numbers of the second kind?

• At least for the usual Stirling numbers of the second kind, Mathematica has these implemented as StirlingS2, see here Jan 16, 2022 at 11:08
• Yes, I know it. Also Mathematica has a code fo q-binomial
– d.y
Jan 16, 2022 at 13:37
• What have you tried? Jan 16, 2022 at 13:49
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• @Hausdorff Thank you so much. But , you write this code based on definition of Matthieu Josuat-Verges paper. our definition for my new problems is according to the Cigler definition. It is possible write the Mathematica code of q-stirling numbers according Ciqler definitions? Thanks in advance
– d.y
Jan 18, 2022 at 12:40

On second thought, it is much more convenient to use Theorem 4.5 of the paper as a definition, so

A[n_, k_, i_, j_] := Binomial[n, k + i] Binomial[n, k - j] -
Binomial[n, k + i + 1] Binomial[n, k - j - 1];

QStirlingS2Fast[n_, k_, q_] :=
1/(1 - q)^(n - k) Sum[(-1)^i*
A[n, k, i, j] q^Binomial[j + 1, 2] QBinomial[i, j, q],
{j, 0,k}, {i, j, n - k}];


This is of course much faster than the implementation via partitions

QStirlingS2[10, 5, 3] // RepeatedTiming
(* {2.00612, 3421737} *)

QStirlingS2Fast[10, 5, 3] // RepeatedTiming
(* {0.000111509, 3421737} *)


and works for large inputs as well

QStirlingS2Fast[1000, 20, 3] // N // RepeatedTiming
(* {0.8409685, 1.465117*^8966} *)


Original post

Here is one possible implementation, which is a bit brute force. I am using Combinatorica  to obtain the set partitions, as well as the definition of the q-Stirling number of the second kind via the number of partition crossings $$cr(\pi)$$ given in the linked paper.

<< Combinatorica

cr[pi_] := Cases[
Flatten[
Tuples /@ Map[
Partition[#, 2, 1] &,
Subsets[Replace[pi, {_} :> Nothing, 1], {2}],
{2}],
1],
{{i_, j_}, {k_, l_}} /; i < k < j < l || k < i < l < j
];

QStirlingS2[n_, k_, q_] := Sum[q^Length@cr[pi], {pi, KSetPartitions[Range@n, k]}]


This has the behavior for $$q\to0$$ given in eq.$$(13)$$,

Narayana[n_, k_] := 1/n*Binomial[n, k - 1] Binomial[n, k];
{ QStirlingS2[10, 4, q] /. q -> 0, Narayana[10, 4] }
(* {2520, 2520} *)


and also satisfies the identity of Theorem 4.5 in your reference

A[n_, k_, i_, j_] := Binomial[n, k + i] Binomial[n, k - j] -
Binomial[n, k + i + 1] Binomial[n, k - j - 1];

Theorem45[n_, k_, q_] :=
{
QStirlingS2[n, k, q] (1 - q)^(n - k),
Sum[(-1)^i*A[n, k, i, j] q^Binomial[j + 1, 2] QBinomial[i, j, q],
{j, 0, k}, {i, j, n - k}]
}

Theorem45[6, 2, 3]
(* {1872, 1872} *)