A274886 Triangle read by rows, the q-analog of the extended Catalan numbers A057977.

1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 2, 2, 2, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 2, 3, 4, 4, 5, 4, 4, 3, 2, 1, 1, 1, 0, 1, 1, 2, 1, 2, 1, 2, 1, 1, 0, 1, 1, 1, 2, 3, 5, 6, 8, 9, 11, 11, 12, 11, 11, 9, 8, 6, 5, 3, 2, 1, 1, 1, 0, 1, 1, 2, 2, 3, 2, 4, 3, 4, 3, 4, 2, 3, 2, 2, 1, 1, 0, 1

Offset: 0

Author

Peter Luschny, Jul 20 2016

Keywords

• nonn
• tabf

Comments

The q-analog of the extended Catalan numbers A057977 are univariate polynomials over the integers with degree floor((n+1)/2)*(floor((n+1)/2)-1)+1.

The q-analog of the Catalan numbers are A129175.

For a combinatorial interpretation in terms of the major index statistic of orbitals see A274888 and the link 'Orbitals'.

Examples

			The polynomials start:
[0] 1
[1] 1
[2] 1
[3] q^2 + q + 1
[4] q^2 + 1
[5] (q^2 + 1) * (q^4 + q^3 + q^2 + q + 1)
[6] (q^2 - q + 1) * (q^4 + q^3 + q^2 + q + 1)
The coefficients of the polynomials are:
[ 0] [1]
[ 1] [1]
[ 2] [1]
[ 3] [1, 1, 1]
[ 4] [1, 0, 1]
[ 5] [1, 1, 2, 2, 2, 1, 1]
[ 6] [1, 0, 1, 1, 1, 0, 1]
[ 7] [1, 1, 2, 3, 4, 4, 5, 4, 4, 3, 2, 1, 1]
[ 8] [1, 0, 1, 1, 2, 1, 2, 1, 2, 1, 1, 0, 1]
[ 9] [1, 1, 2, 3, 5, 6, 8, 9, 11, 11, 12, 11, 11, 9, 8, 6, 5, 3, 2, 1, 1]
[10] [1, 0, 1, 1, 2, 2, 3, 2, 4, 3, 4, 3, 4, 2, 3, 2, 2, 1, 1, 0, 1]
		

Links

• Peter Luschny, Orbitals

Crossrefs

Cf. A057977, A129175, A274885, A274888.

Programs

Maple

QExtCatalan := proc(n) local h, p, P;
P := x -> QDifferenceEquations:-QPochhammer(q,q,x);
h := iquo(n, 2): p := `if`(n::even, 1-q, 1); (p*P(n))/(P(h)*P(h+1));
expand(simplify(expand(%))); seq(coeff(%, q, j), j=0..degree(%)) end:
seq(QExtCatalan(n, q), n=0..10);

Mathematica

(* Function QBinom1 is defined in A274885. *)
QExtCatalan[n_] := QBinom1[n] / QBinomial[n+1,1,q]; Table[CoefficientList[ QExtCatalan[n] // FunctionExpand,q], {n,0,10}] // Flatten

Sage

# uses[q_binom1 from A274885]
from sage.combinat.q_analogues import q_int
def q_ext_catalan_number(n): return q_binom1(n)//q_int(n+1)
for n in (0..10): print([n], q_ext_catalan_number(n).list())

Sage

# uses[unit_orbitals from A274709]
# Brute force counting
def catalan_major_index(n):
    S = [0]*(((n+1)//2)^2 + ((n+1) % 2) - (n//2))
    for u in unit_orbitals(n):
        if any(x > 0 for x in accumulate(u)): continue # never rise above 0
        L = [i+1 if u[i+1] < u[i] else 0 for i in (0..n-2)]
        #    i+1 because u is 0-based whereas convention assumes 1-base.
        S[sum(L)] += 1
    return S
for n in (0..10): print(catalan_major_index(n))

Formula

q-extCatalan(n,q) = (p*P(n,q))/(P(h,q)*P(h+1,q)) with P(n,q) = q-Pochhammer(n,q), h = floor(n/2) and p = 1-q if n is even else 1.