A275332 Triangle read by rows: the major index statistic of the oscillating orbitals, also the q-analog of the oscillating orbitals A232500.

0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 2, 2, 2, 1, 1, 0, 1, 1, 1, 2, 2, 1, 1, 1, 0, 1, 2, 3, 5, 7, 8, 9, 9, 8, 7, 5, 3, 2, 1, 0, 1, 1, 2, 2, 4, 4, 5, 4, 5, 4, 4, 2, 2, 1, 1, 0, 1, 2, 4, 6, 10, 14, 19, 23, 28, 31, 34, 34, 34, 31, 28, 23, 19, 14, 10, 6, 4, 2, 1

Offset

0,12

Comments

The q-osc_orbitals are univariate polynomials over the integers with degree floor((n+1)/2)^2 - n mod 2. Evaluated at q=1 they give the oscillating orbitals A232500(n) for n>=2.

Combinatorial interpretation: The definition of an orbital system is given in A232500 and in the link 'Orbitals'. The major index of an orbital is the sum of the positions of steps which are immediately followed by a step with strictly smaller value. The major index of the oscillating orbitals is the restriction of the major index of all orbitals (see A274888) to this subclass.

Links

Table of n, a(n) for n=0..79.

Peter Luschny, Orbitals

Formula

Let A(n,q) = q*alpha(n-2,q)/alpha(n,q) with alpha(n,q) = Sum_{j=0..n} q^j and B(n,q) = q*beta(n-1,q)/beta(n,q) with beta(n,q) = Sum_{j=0..n} q^(2*j). Then QOscOrbitals(n,q) = qSwing(n,q)*C(n,q) with C(n,q) = A(floor(n/2),q) if n mod 4 in {0, 1} else C(n,q) = B(floor(n/4),q).

Example

Some polynomials:
[4] (q^2 + 1)*q
[5] (q^4 + q^3 + q^2 + q + 1)*(q^2 + 1)*q
[6] (q^4 + q^3 + q^2 + q + 1)*(q^2 - q + 1)*(q + 1)*q
[7] (q^6 + q^5 + q^4 + q^3 + q^2 + q + 1)*(q^4 + q^3 + q^2 + q + 1)*(q^2 - q + 1)*(q + 1)*q
[8] (q^6 + q^5 + q^4 + q^3 + q^2 + q + 1)*(q^4 + 1)*(q^2 + q + 1)*(q^2 - q + 1)*q
[9] (q^6 + q^5 + q^4 + q^3 + q^2 + q + 1)*(q^6 + q^3 + 1)*(q^4 + 1)*(q^2 + q + 1)^2*(q^2 - q + 1)*q
The triangle starts:
[n] [k=0,1,2,...] [row sum]
[0] [0] 0
[1] [0] 0
[2] [0] 0
[3] [0] 0
[4] [0, 1, 0, 1] 2
[5] [0, 1, 1, 2, 2, 2, 1, 1] 10
[6] [0, 1, 1, 1, 2, 2, 1, 1, 1] 10
[7] [0, 1, 2, 3, 5, 7, 8, 9, 9, 8, 7, 5, 3, 2, 1] 70
[8] [0, 1, 1, 2, 2, 4, 4, 5, 4, 5, 4, 4, 2, 2, 1, 1] 42
T(5,3) = 2 because A = [-1, 1, 1, -1, 0] and B = [1, 0, -1, -1, 1] are oscillating orbitals; A has downsteps at position 3 and B has downsteps at positions 1 and 2.

Prog

(Sage)
from sage.combinat.q_analogues import q_factorial
def osc_orbitals_coeffs(n):
    q = var('q')
    if n < 4: return [0]
    a = lambda n, q: sum(q^j for j in (0..n))
    b = lambda n, q: sum(q^(2*j) for j in (0..n))
    A = lambda n, q: q*a(n-2, q)/a(n, q)
    B = lambda n, q: q*b(n-1, q)/b(n, q)
    Q = A(n//2, q) if n%4 == 0 or n%4 == 1 else B(n//4, q)
    qSwing = lambda n, q: q_factorial(n, q)/q_factorial(n//2, q)^2
    return ((Q*qSwing(n, q)).factor()).list()
for n in (0..10): print([n], osc_orbitals_coeffs(n))
(Sage) # uses[unit_orbitals from A274709]
# Brute force counting
def osc_orbitals_major_index(n):
    if n<4: return [0]
    S = [0]*(((n+1)//2)^2 - (n % 2))
    for u in unit_orbitals(n):
        if all(x >= 0 for x in accumulate(u)): continue
        if all(x <= 0 for x in accumulate(u)): continue
        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(osc_orbitals_major_index(n))

Crossrefs

Cf. A056040 (row sums), A274887 (q-factorial), A274888 (q-swinging factorial),

A274884 (alternate description of oscillating orbitals).

Sequence in context: A029446 A358479 A288160 * A029442 A125917 A071468

Adjacent sequences: A275329 A275330 A275331 * A275333 A275334 A275335

Keyword

nonn,tabf

Author

Peter Luschny, Jul 26 2016

Status

approved