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

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

Author

Peter Luschny, Jul 10 2016

Keywords

• nonn
• tabf

Comments

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.

Examples

			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].
		

Links

• Peter Luschny, Orbitals

Crossrefs

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).

Programs

Sage

# 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))