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

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

Author

Peter Luschny, Jul 12 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 ascent of an orbital is its longest up-run.

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, 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].
		

Links

• Peter Luschny, Orbitals

Crossrefs

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).

Programs

Sage

# 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))