A232500 -id:A232500 - cp's OEIS frontend

A274884 Triangle read by rows, coefficients of q-polynomials representing the oscillating orbitals over n sectors as A274888(n) - 2*A274886(n), a q-analog of A232500.

-1, -1, -1, 1, -1, 0, 0, 1, -1, 1, 0, 1, 1, -1, 0, 0, 1, 2, 3, 2, 2, 1, -1, 1, 0, 1, 1, 3, 1, 2, 1, 1, -1, 0, 0, 1, 2, 5, 6, 9, 9, 10, 9, 8, 5, 4, 2, 1, -1, 1, 0, 1, 1, 3, 3, 5, 4, 5, 5, 5, 3, 3, 2, 1, 1

Offset: 0

Peter Luschny, Jul 20 2016

The polynomials are univariate polynomials over the integers with degree floor((n+1)/2)^2 + ((n+1) mod 2). Evaluated at q=1 the polynomials give A232500.

For the combinatorial interpretation see A232500 and the link 'orbitals' (see also the illustrations there).

			The polynomials start:
[0] -1
[1] -1
[2] q - 1
[3] (q - 1) * (q^2 + q + 1)
[4] (q^2 + 1) * (q^2 + q - 1)
[5] (q^2 + 1) * (q^2 + q - 1) * (q^4 + q^3 + q^2 + q + 1)
[6] (q^2 - q + 1) * (q^3 + q^2 + q - 1) * (q^4 + q^3 + q^2 + q + 1)
The table starts:
[n] [k=0,1,2,...] [row sum]
[0] [-1] -1
[1] [-1] -1
[2] [-1, 1] 0
[3] [-1, 0, 0, 1] 0
[4] [-1, 1, 0, 1, 1] 2
[5] [-1, 0, 0, 1, 2, 3, 2, 2, 1] 10
[6] [-1, 1, 0, 1, 1, 3, 1, 2, 1, 1] 10
[7] [-1, 0, 0, 1, 2, 5, 6, 9, 9, 10, 9, 8, 5, 4, 2, 1] 70
[8] [-1, 1, 0, 1, 1, 3, 3, 5, 4, 5, 5, 5, 3, 3, 2, 1, 1] 42

Peter Luschny, Orbitals

Cf. A232500 (row sums), A274886, A274888.

QOscOrbitals := proc(n) local h, p, P, F, C, S;
P := x -> QDifferenceEquations:-QPochhammer(q,q,x);
F := x -> QDifferenceEquations:-QFactorial(x,q);
h := iquo(n,2): p := `if`(n::even,1-q,1);
C := (p*P(n))/(P(h)*P(h+1)); S := F(n)/F(h)^2;
expand(simplify(expand(S-2*C))); seq(coeff(%,q,j), j=0..degree(%)) end:
seq(QOscOrbitals(n), n=0..8);

# uses[q_ext_catalan_number]
# Function q_ext_catalan_number is in A274886.
from sage.combinat.q_analogues import q_multinomial
def q_osc_orbitals(n):
    return q_multinomial([n//2, n%2, n//2]) - 2*q_ext_catalan_number(n)
for n in (0..9): print(q_osc_orbitals(n).list())

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

Original entry on oeis.org

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

Peter Luschny, Jul 26 2016

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.

			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.

Peter Luschny, Orbitals

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

A274884 (alternate description of oscillating orbitals).

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))

# 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))

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).

A241477 Triangle read by rows, number of orbitals classified with respect to the first zero crossing, n>=1, 1<=k<=n.

Original entry on oeis.org

1, 0, 2, 2, 2, 2, 0, 4, 0, 2, 6, 12, 4, 2, 6, 0, 12, 0, 4, 0, 4, 20, 60, 12, 12, 12, 4, 20, 0, 40, 0, 12, 0, 8, 0, 10, 70, 280, 40, 60, 36, 24, 40, 10, 70, 0, 140, 0, 40, 0, 24, 0, 20, 0, 28, 252, 1260, 140, 280, 120, 120, 120, 60, 140, 28, 252, 0, 504, 0

Offset: 1

Peter Luschny, Apr 23 2014

For the combinatorial definitions see A232500. An orbital w over n sectors has its first zero crossing at k if k is the smallest j such that the partial sum(1<=i<=j, w(i))) = 0, where w(i) are the jumps of the orbital represented by -1, 0, 1.

			[1], [ 1]
[2], [ 0,  2]
[3], [ 2,  2,  2]
[4], [ 0,  4,  0,  2]
[5], [ 6, 12,  4,  2,  6]
[6], [ 0, 12,  0,  4,  0, 4]
[7], [20, 60, 12, 12, 12, 4, 20]

Row sums: A056040.

Cf. A232500.

A241477 := proc(n, k)
  if n = 0 then 1
elif k = 0 then 0
elif irem(n, 2) = 0 and irem(k, 2) = 1 then 0
elif k = 1 then (n-1)!/iquo(n-1,2)!^2
else 2*(n-k)!*(k-2)!/iquo(k,2)/(iquo(k-2,2)!*iquo(n-k,2)!)^2
  fi end:
for n from 1 to 9 do seq(A241477(n, k), k=1..n) od;

T[n_, k_] := Which[n == 0, 1, k == 0, 0, Mod[n, 2] == 0 && Mod[k, 2] == 1,  0, k == 1, (n-1)!/Quotient[n-1, 2]!^2, True, 2*(n-k)!*(k-2)!/Quotient[k, 2]/(Quotient[k-2, 2]!*Quotient[n-k, 2]!)^2];
Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 20 2018, from Maple *)

def A241477_row(n):
    if n == 0: return [1]
    Z = [0]*n; T = [0] if is_odd(n) else []
    for i in (1..n//2): T.append(-1); T.append(1)
    for p in Permutations(T):
        i = 0; s = p[0]
        while s != 0: i += 1; s += p[i];
        Z[i] += 1
    return Z
for n in (1..9): A241477_row(n)

If n is even and k is odd then T(n, k) = 0 else if k = 1 then T(n, 1) = A056040(n-1) else T(n, k) = 2*A057977(k-2)*A056040(n-k).
T(n, n) = A241543(n).
T(n+1, 1) = A126869(n).
T(2*n, 2*n) = |A002420(n)|.
T(2*n+1, 1) = A000984(n).
T(2*n+1, n+1) = A241530(n).
T(2*n+2, 2) = A028329(n).
T(4*n, 2*n) = |A010370(n)|.
T(4*n, 4*n) = |A024491(n)|.
T(4*n+1, 1) = A001448(n).
T(4*n+1, 2*n+1) = A002894(n).

A274709 A statistic on orbital systems over n sectors: the number of orbitals which rise to maximum height k over the central circle.

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 2, 3, 1, 10, 15, 5, 5, 9, 5, 1, 35, 63, 35, 7, 14, 28, 20, 7, 1, 126, 252, 180, 63, 9, 42, 90, 75, 35, 9, 1, 462, 990, 825, 385, 99, 11, 132, 297, 275, 154, 54, 11, 1, 1716, 3861, 3575, 2002, 702, 143, 13, 429, 1001, 1001, 637, 273, 77, 13, 1

Offset: 0

Peter Luschny, Jul 09 2016

The definition of an orbital system is given in A232500 (see also the illustration there). The number of orbitals over n sectors is counted by the swinging factorial A056040.
Note that (sum row_n) / row_n(0) = 1,1,2,2,3,3,4,4,..., i.e. the swinging factorials are multiples of the extended Catalan numbers A057977 generalizing the fact that the central binomials are multiples of the Catalan numbers.
T(n, k) is a subtriangle of the extended Catalan triangle A189231.

			Triangle read by rows, n>=0. The length of row n is floor((n+2)/2).
[ n] [k=0,1,2,...] [row sum]
[ 0] [  1] 1
[ 1] [  1] 1
[ 2] [  1,   1] 2
[ 3] [  3,   3] 6
[ 4] [  2,   3,   1] 6
[ 5] [ 10,  15,   5] 30
[ 6] [  5,   9,   5,   1] 20
[ 7] [ 35,  63,  35,   7] 140
[ 8] [ 14,  28,  20,   7,  1] 70
[ 9] [126, 252, 180,  63,  9] 630
[10] [ 42,  90,  75,  35,  9,  1] 252
[11] [462, 990, 825, 385, 99, 11] 2772
[12] [132, 297, 275, 154, 54, 11, 1] 924
T(6, 2) = 5 because the five orbitals [-1, 1, 1, 1, -1, -1], [1, -1, 1, 1, -1, -1], [1, 1, -1, -1, -1, 1], [1, 1, -1, -1, 1, -1], [1, 1, -1, 1, -1, -1] raise to maximal height of 2 over the central circle.

Peter Luschny, The lost Catalan numbers
Peter Luschny, Orbitals

Cf. A008313, A039599 (even rows), A047072, A056040 (row sums), A057977 (col 0), A063549 (col 0), A112467, A120730, A189230 (odd rows aerated), A189231, A232500.

Other orbital statistics: A241477 (first zero crossing), A274706 (absolute integral), A274708 (number of peaks), A274710 (number of turns), A274878 (span), A274879 (returns), A274880 (restarts), A274881 (ascent).

S := proc(n,k) option remember; `if`(k>n or k<0, 0, `if`(n=k, 1, S(n-1,k-1)+
modp(n-k,2)*S(n-1,k)+S(n-1,k+1))) end: T := (n,k) -> S(n,2*k);
seq(print(seq(T(n,k), k=0..iquo(n,2))), n=0..12);

from itertools import accumulate
# Brute force counting
def unit_orbitals(n):
    sym_range = [i for i in range(-n+1, n, 2)]
    for c in Combinations(sym_range, n):
        P = Permutations([sgn(v) for v in c])
        for p in P: yield p
def max_orbitals(n):
    if n == 0: return [1]
    S = [0]*((n+2)//2)
    for u in unit_orbitals(n):
        L = list(accumulate(u))
        S[max(L)] += 1
    return S
for n in (0..10): print(max_orbitals(n))

A274706 Irregular triangle read by rows. T(n,k) (n >= 0) is a statistic on orbital systems over n sectors: the number of orbitals which have an integral whose absolute value is k.

Original entry on oeis.org

1, 1, 0, 2, 0, 4, 2, 2, 0, 2, 0, 2, 6, 4, 6, 4, 4, 4, 2, 0, 6, 0, 6, 0, 4, 0, 2, 0, 2, 6, 24, 16, 20, 14, 16, 12, 8, 6, 8, 4, 4, 2, 8, 0, 14, 0, 14, 0, 10, 0, 10, 0, 6, 0, 4, 0, 2, 0, 2, 36, 52, 68, 48, 64, 48, 48, 40, 44, 32, 36, 24, 22, 16, 16, 8, 10, 8, 4, 4, 2

Offset: 0

Peter Luschny, Jul 10 2016

For the combinatorial definitions see A232500. The absolute integral of an orbital w over n sectors is abs(Sum_{k=1..n} Sum_{i=1..k} w(i)) where w(i) are the jumps of the orbital represented by -1, 0, 1.

An orbital is balanced if its integral is 0 (A241810).

			The length of row n is 1+floor(n^2//4).
The triangle begins:
  [n] [k=0,1,2,...] [row sum]
  [0] [1] 1
  [1] [1] 1
  [2] [0, 2] 2
  [3] [0, 4, 2] 6
  [4] [2, 0, 2, 0, 2] 6
  [5] [6, 4, 6, 4, 4, 4, 2] 30
  [6] [0, 6, 0, 6, 0, 4, 0, 2, 0, 2] 20
  [7] [6, 24, 16, 20, 14, 16, 12, 8, 6, 8, 4, 4, 2] 140
  [8] [8, 0, 14, 0, 14, 0, 10, 0, 10, 0, 6, 0, 4, 0, 2, 0, 2] 70
T(5, 4) = 4 because the integral of four orbitals have the absolute value 4:
  Integral([-1, -1, 1, 1, 0]) = -4, Integral([0, -1, -1, 1, 1]) = -4,
  Integral([0, 1, 1, -1, -1]) = 4, Integral([1, 1, -1, -1, 0]) = 4.

Peter Luschny, Orbitals

Cf. A056040 (row sum), A232500, A241810 (col. 0), A242087.

Other orbital statistics: A241477 (first zero crossing), A274708 (number of peaks), A274709 (max. height), A274710 (number of turns), A274878 (span), A274879 (returns), A274880 (restarts), A274881 (ascent).

from itertools import accumulate
# Brute force counting
def unit_orbitals(n):
    sym_range = [i for i in range(-n+1, n, 2)]
    for c in Combinations(sym_range, n):
        P = Permutations([sgn(v) for v in c])
        for p in P: yield p
def orbital_integral(n):
    if n == 0: return [1]
    S = [0]*(1+floor(n^2//4))
    for u in unit_orbitals(n):
        L = list(accumulate(accumulate(u)))
        S[abs(L[-1])] += 1
    return S
for n in (0..8): print(orbital_integral(n))

A274710 A statistic on orbital systems over n sectors: the number of orbitals which make k turns.

Original entry on oeis.org

1, 1, 0, 2, 0, 0, 6, 0, 2, 2, 2, 0, 0, 6, 12, 12, 0, 2, 4, 8, 4, 2, 0, 0, 6, 24, 52, 40, 18, 0, 2, 6, 18, 18, 18, 6, 2, 0, 0, 6, 36, 120, 180, 180, 84, 24, 0, 2, 8, 32, 48, 72, 48, 32, 8, 2, 0, 0, 6, 48, 216, 480, 744, 672, 432, 144, 30, 0, 2, 10, 50, 100, 200, 200, 200, 100, 50, 10, 2

Offset: 0

Peter Luschny, Jul 10 2016

The definition of an orbital system is given in A232500 (see also the illustration there). The number of orbitals over n sectors is counted by the swinging factorial A056040.
A 'turn' of an orbital w takes place where signum(w[i]) is not equal to signum(w[i+1]).
A152659 is a subtriangle.

			Triangle read by rows, n>=0. The length of row n is n for n>=1.
[n] [k=0,1,2,...]                      [row sum]
[0] [1]                                    1
[1] [1]                                    1
[2] [0, 2]                                 2
[3] [0, 0, 6]                              6
[4] [0, 2, 2,  2]                          6
[5] [0, 0, 6, 12,  12]                    30
[6] [0, 2, 4,  8,   4,   2]               20
[7] [0, 0, 6, 24,  52,  40,  18]         140
[8] [0, 2, 6, 18,  18,  18,   6,  2]      70
[9] [0, 0, 6, 36, 120, 180, 180, 84, 24] 630
T(5,2) = 6 because the six orbitals [-1, -1, 0, 1, 1], [-1, -1, 1, 1, 0], [0, -1, -1, 1, 1], [0, 1, 1, -1, -1], [1, 1, -1, -1, 0], [1, 1, 0, -1, -1] make 2 turns.

Peter Luschny, Orbitals

Cf. A056040 (row sum), A152659, A232500.

Other orbital statistics: A241477 (first zero crossing), A274706 (absolute integral), A274708 (number of peaks), A274709 (max. height), A274878 (span), A274879 (returns), A274880 (restarts), A274881 (ascent).

# uses[unit_orbitals from A274709]
# Brute force counting
def orbital_turns(n):
    if n == 0: return [1]
    S = [0]*(n)
    for u in unit_orbitals(n):
        L = sum(0 if sgn(u[i]) == sgn(u[i+1]) else 1 for i in (0..n-2))
        S[L] += 1
    return S
for n in (0..12): print(orbital_turns(n))

For even n>0: T(n,k) = 2*C(n/2-1,(k-1+mod(k-1,2))/2)*C(n/2-1,(k-1-mod(k-1,2))/2) for k=0..n-1 (from A152659).

A274878 A statistic on orbital systems over n sectors: the number of orbitals with span k.

Original entry on oeis.org

1, 1, 0, 2, 0, 6, 0, 2, 4, 0, 10, 20, 0, 2, 12, 6, 0, 14, 84, 42, 0, 2, 28, 32, 8, 0, 18, 252, 288, 72, 0, 2, 60, 120, 60, 10, 0, 22, 660, 1320, 660, 110, 0, 2, 124, 390, 300, 96, 12, 0, 26, 1612, 5070, 3900, 1248, 156, 0, 2, 252, 1176, 1260, 588, 140, 14

Offset: 0

Peter Luschny, Jul 10 2016

The definition of an orbital system is given in A232500 (see also the illustration there). The number of orbitals over n sectors is counted by the swinging factorial A056040.

The 'span' of an orbital w is the difference between the highest and the lowest level of the orbital system touched by w.

			Triangle read by rows, n>=0. The length of row n is floor((n+2)/2).
[ n] [k=0,1,2,...]          [row sum]
[ 0] [1]                        1
[ 1] [1]                        1
[ 2] [0, 2]                     2
[ 3] [0, 6]                     6
[ 4] [0, 2, 4]                  6
[ 5] [0, 10, 20]               30
[ 6] [0, 2, 12, 6]             20
[ 7] [0, 14, 84, 42]          140
[ 8] [0, 2, 28, 32, 8]         70
[ 9] [0, 18, 252, 288, 72]    630
[10] [0, 2, 60, 120, 60, 10]  252
T(6, 3) = 6 because the span of the following six orbitals is 3:
[-1, -1, -1, 1, 1, 1], [-1, -1, 1, 1, 1, -1], [-1, 1, 1, 1, -1, -1],
[1, -1, -1, -1, 1, 1], [1, 1, -1, -1, -1, 1], [1, 1, 1, -1, -1, -1].

Peter Luschny, Orbitals

Cf. A056040 (row sum), A232500.

Other orbital statistics: A241477 (first zero crossing), A274706 (absolute integral), A274708 (number of peaks), A274709 (max. height), A274710 (number of turns), A274879 (returns), A274880 (restarts), A274881 (ascent).

# uses[unit_orbitals from A274709]
from itertools import accumulate
# Brute force counting.
def orbital_span(n):
    if n == 0: return [1]
    S = [0]*((n+2)//2)
    for u in unit_orbitals(n):
        L = list(accumulate(u))
        S[max(L) - min(L)] += 1
    return S
for n in (0..10): print(orbital_span(n))

A274879 A statistic on orbital systems over n sectors: the number of orbitals with k returns.

Original entry on oeis.org

1, 1, 2, 2, 4, 2, 4, 6, 12, 12, 4, 8, 8, 20, 40, 48, 32, 10, 20, 24, 16, 70, 140, 180, 160, 80, 28, 56, 72, 64, 32, 252, 504, 672, 672, 480, 192, 84, 168, 224, 224, 160, 64, 924, 1848, 2520, 2688, 2240, 1344, 448, 264, 528, 720, 768, 640, 384, 128

Offset: 0

Peter Luschny, Jul 11 2016

The definition of an orbital system is given in A232500 (see also the illustration there). The number of orbitals over n sectors is counted by the swinging factorial A056040.
When a segment of an orbital starts at a point on the central circle this point is called a 'return' of the orbital if it is not the origin.
If an orbital touches the central circle only in the origin it is called a prime orbital. Column 0 counts the prime orbitals over n sectors.
A108747 is a subtriangle.

			Triangle read by rows, n>=0. The length of row n is floor((n+1)/2) for n>=1.
[ n] [k=0,1,2,...] [row sum]
[ 0] [1] 1
[ 1] [1] 1
[ 2] [2] 2
[ 3] [2, 4] 6
[ 4] [2, 4] 6
[ 5] [6, 12, 12] 30
[ 6] [4, 8, 8] 20
[ 7] [20, 40, 48, 32] 140
[ 8] [10, 20, 24, 16] 70
[ 9] [70, 140, 180, 160, 80] 630
[10] [28, 56, 72, 64, 32] 252
[11] [252, 504, 672, 672, 480, 192] 2772
T(6,0) = 4 because the following 4 orbitals stay above or below the central
circle: [-1, -1, -1, 1, 1, 1], [-1, -1, 1, -1, 1, 1], [1, 1, -1, 1, -1, -1],
[1, 1, 1, -1, -1, -1].

Peter Luschny, Orbitals

Cf. A056040 (row sum), A108747, A232500, A241543 (col. 0).

Other orbital statistics: A241477 (first zero crossing), A274706 (absolute integral), A274708 (peaks), A274709 (max. height), A274710 (number of turns), A274878 (span), A274880 (restarts), A274881 (ascent).

# uses[unit_orbitals from A274709]
from itertools import accumulate
# Brute force counting
def orbital_returns(n):
    if n == 0: return [1]
    S = [0]*((n+1)//2)
    for u in unit_orbitals(n):
        L = list(accumulate(u))
        Z = len(list(filter(lambda z: z == 0, L)))
        S[Z-1] += 1  # exclude origin
    return S
for n in (0..10): print(orbital_returns(n))

For even n>0: T(n,k) = 2^(k+1)*(k+1)*binomial(n-k-1,n/2)/(n-k-1) for k=0..n/2-1 (from A108747).

A274880 A statistic on orbital systems over n sectors: the number of orbitals with k restarts.

Original entry on oeis.org

1, 1, 2, 5, 1, 4, 2, 18, 11, 1, 10, 8, 2, 65, 57, 17, 1, 28, 28, 12, 2, 238, 252, 116, 23, 1, 84, 96, 54, 16, 2, 882, 1050, 615, 195, 29, 1, 264, 330, 220, 88, 20, 2, 3300, 4257, 2915, 1210, 294, 35, 1, 858, 1144, 858, 416, 130, 24, 2, 12441, 17017, 13013, 6461, 2093, 413, 41, 1

Offset: 0

Peter Luschny, Jul 11 2016

The definition of an orbital system is given in A232500 (see also the illustration there). The number of orbitals over n sectors is counted by the swinging factorial A056040.
A 'restart' of an orbital is a raise which starts from the central circle.
A118920 is a subtriangle.

			Triangle read by rows, n>=0. The length of row n is floor((n+1)/2) for n>=1.
[n] [k=0,1,2,...]                 [row sum]
[ 0] [1]                              1
[ 1] [1]                              1
[ 2] [2]                              2
[ 3] [5, 1]                           6
[ 4] [4, 2]                           6
[ 5] [18, 11, 1]                     30
[ 6] [10, 8, 2]                      20
[ 7] [65, 57, 17, 1]                140
[ 8] [28, 28, 12, 2]                 70
[ 9] [238, 252, 116, 23, 1]         630
[10] [84, 96, 54, 16, 2]            252
[11] [882, 1050, 615, 195, 29, 1]  2772
T(6, 2) = 2 because there are two orbitals over 6 segments which have 2 ascents:
[-1, 1, 1, -1, 1, -1] and [1, -1, 1, -1, 1, -1].

Peter Luschny, Orbitals

Cf. A056040 (row sum), A118920, A232500.

Other orbital statistics: A241477 (first zero crossing), A274706 (absolute integral), A274708 (peaks), A274709 (max. height), A274710 (number of turns), A274878 (span), A274879 (returns), A274881 (ascent).

# uses[unit_orbitals from A274709]
from itertools import accumulate
# Brute force counting
def orbital_restart(n):
    if n == 0: return [1]
    S = [0]*((n+1)//2)
    for u in unit_orbitals(n):
        A = list(accumulate(u))
        L = [1 if A[i] == 0 and A[i+1] == 1  else 0 for i in (0..n-2)]
        S[sum(L)] += 1
    return S
for n in (0..12): print(orbital_restart(n))

For even n>0: T(n,k) = 4*(k+1)*binomial(n,n/2-k-1)/n for k=0..n/2-1 (from A118920).

A274881 A statistic on orbital systems over n sectors: the number of orbitals which have an ascent of length k.

Original entry on oeis.org

1, 1, 0, 2, 0, 6, 0, 3, 3, 0, 18, 12, 0, 4, 12, 4, 0, 40, 80, 20, 0, 5, 40, 20, 5, 0, 75, 375, 150, 30, 0, 6, 120, 90, 30, 6, 0, 126, 1470, 882, 252, 42, 0, 7, 350, 371, 147, 42, 7, 0, 196, 5292, 4508, 1568, 392, 56, 0, 8, 1008, 1456, 672, 224, 56, 8

Offset: 0

Peter Luschny, Jul 12 2016

The definition of an orbital system is given in A232500 (see also the illustration there). The number of orbitals over n sectors is counted by the swinging factorial A056040.

The ascent of an orbital is its longest up-run.

			Triangle read by rows, n>=0. The length of row n is floor((n+2)/2).
[ n] [k=0,1,2,...]                [row sum]
[ 0] [1]                              1
[ 1] [1]                              1
[ 2] [0, 2]                           2
[ 3] [0, 6]                           6
[ 4] [0, 3, 3]                        6
[ 5] [0, 18, 12]                     30
[ 6] [0, 4, 12, 4]                   20
[ 7] [0, 40, 80, 20]                140
[ 8] [0, 5, 40, 20, 5]               70
[ 9] [0, 75, 375, 150, 30]          630
[10] [0, 6, 120, 90, 30, 6]         252
[11] [0, 126, 1470, 882, 252, 42]  2772
[12] [0, 7, 350, 371, 147, 42, 7]   924
T(6,3) = 4 because four orbitals over six sectors have a maximal up-run of length 3.
[-1,-1,-1,1,1,1], [-1,-1,1,1,1,-1], [-1,1,1,1,-1,-1], [1,1,1,-1,-1,-1].

Peter Luschny, Orbitals

Cf. A056040 (row sum), A232500.

Other orbital statistics: A241477 (first zero crossing), A274706 (absolute integral), A274708 (peaks), A274709 (max. height), A274710 (number of turns), A274878 (span), A274879 (returns), A274880 (restarts).

# uses[unit_orbitals from A274709]
# Brute force counting
def orbital_ascent(n):
    if n < 2: return [1]
    S = [0]*((n+2)//2)
    for u in unit_orbitals(n):
        B = [0]*n
        for i in (0..n-1):
            B[i] = 0 if u[i] <= 0 else B[i-1] + u[i]
        S[max(B)] += 1
    return S
for n in (0..12): print(orbital_ascent(n))

cp's OEIS Frontend

A274884 Triangle read by rows, coefficients of q-polynomials representing the oscillating orbitals over n sectors as A274888(n) - 2*A274886(n), a q-analog of A232500.

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

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A241477 Triangle read by rows, number of orbitals classified with respect to the first zero crossing, n>=1, 1<=k<=n.

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A274709 A statistic on orbital systems over n sectors: the number of orbitals which rise to maximum height k over the central circle.

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A274706 Irregular triangle read by rows. T(n,k) (n >= 0) is a statistic on orbital systems over n sectors: the number of orbitals which have an integral whose absolute value is k.

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A274710 A statistic on orbital systems over n sectors: the number of orbitals which make k turns.

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A274878 A statistic on orbital systems over n sectors: the number of orbitals with span k.

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A274879 A statistic on orbital systems over n sectors: the number of orbitals with k returns.

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