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A274886 Triangle read by rows, the q-analog of the extended Catalan numbers A057977. 4
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 (list; graph; refs; listen; history; text; internal format)
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)

# The function q_binom1 is defined in 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()

# Brute force counting, function unit_orbitals defined in A274709.

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 * A004571 A204429 A292560

Adjacent sequences:  A274883 A274884 A274885 * A274887 A274888 A274889

KEYWORD

nonn,tabf

AUTHOR

Peter Luschny, Jul 20 2016

STATUS

approved

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Last modified May 27 06:24 EDT 2019. Contains 323599 sequences. (Running on oeis4.)