A275332 - OEIS

%I #14 Mar 27 2020 12:11:30

%S 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,

%T 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,

%U 31,34,34,34,31,28,23,19,14,10,6,4,2,1

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

%C 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.

%C 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.

%H Peter Luschny, <a href="https://oeis.org/wiki/User:Peter_Luschny/Orbitals">Orbitals</a>

%F 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).

%e Some polynomials:

%e [4] (q^2 + 1)*q

%e [5] (q^4 + q^3 + q^2 + q + 1)*(q^2 + 1)*q

%e [6] (q^4 + q^3 + q^2 + q + 1)*(q^2 - q + 1)*(q + 1)*q

%e [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

%e [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

%e [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

%e The triangle starts:

%e [n] [k=0,1,2,...] [row sum]

%e [0] [0] 0

%e [1] [0] 0

%e [2] [0] 0

%e [3] [0] 0

%e [4] [0, 1, 0, 1] 2

%e [5] [0, 1, 1, 2, 2, 2, 1, 1] 10

%e [6] [0, 1, 1, 1, 2, 2, 1, 1, 1] 10

%e [7] [0, 1, 2, 3, 5, 7, 8, 9, 9, 8, 7, 5, 3, 2, 1] 70

%e [8] [0, 1, 1, 2, 2, 4, 4, 5, 4, 5, 4, 4, 2, 2, 1, 1] 42

%e 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.

%o (Sage)

%o from sage.combinat.q_analogues import q_factorial

%o def osc_orbitals_coeffs(n):

%o     q = var('q')

%o     if n < 4: return [0]

%o     a = lambda n,q: sum(q^j for j in (0..n))

%o     b = lambda n,q: sum(q^(2*j) for j in (0..n))

%o     A = lambda n,q: q*a(n-2,q)/a(n,q)

%o     B = lambda n,q: q*b(n-1,q)/b(n,q)

%o     Q = A(n//2,q) if n%4 == 0 or n%4 == 1 else B(n//4,q)

%o     qSwing = lambda n,q: q_factorial(n,q)/q_factorial(n//2,q)^2

%o     return ((Q*qSwing(n,q)).factor()).list()

%o for n in (0..10): print([n], osc_orbitals_coeffs(n))

%o (Sage) # uses[unit_orbitals from A274709]

%o # Brute force counting

%o def osc_orbitals_major_index(n):

%o     if n<4: return [0]

%o     S = [0]*(((n+1)//2)^2 - (n % 2))

%o     for u in unit_orbitals(n):

%o         if all(x >= 0 for x in accumulate(u)): continue

%o         if all(x <= 0 for x in accumulate(u)): continue

%o         L = [i+1 if u[i+1] < u[i] else 0 for i in (0..n-2)]

%o         #    i+1 because u is 0-based whereas convention assumes 1-base

%o         S[sum(L)] += 1

%o     return S

%o for n in (0..10):  print(osc_orbitals_major_index(n))

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

%Y A274884 (alternate description of oscillating orbitals).

%K nonn,tabf

%O 0,12

%A _Peter Luschny_, Jul 26 2016