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A274881
A statistic on orbital systems over n sectors: the number of orbitals which have an ascent of length k.
10
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
0,4
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.
EXAMPLE
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].
PROG
(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))
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).
Sequence in context: A274878 A050821 A076257 * A303638 A162974 A275325
KEYWORD
nonn,tabf
AUTHOR
Peter Luschny, Jul 12 2016
STATUS
approved