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

0,12

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'.

Links

Table of n, a(n) for n=0..90.

Peter Luschny, Orbitals

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.

Example

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]

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

Prog

(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))

Crossrefs

Cf. A057977, A129175, A274885, A274888.

Sequence in context: A029442 A125917 A071468 * A325955 A337311 A372924

Adjacent sequences: A274883 A274884 A274885 * A274887 A274888 A274889

Keyword

nonn,tabf

Author

Peter Luschny, Jul 20 2016

Status

approved