

A275333


Triangle read by rows, the break statistic on orbital systems over n sectors.


0



1, 1, 1, 1, 2, 2, 2, 1, 1, 2, 1, 1, 3, 3, 6, 6, 6, 3, 3, 1, 1, 2, 3, 3, 3, 3, 2, 1, 1, 4, 4, 8, 12, 16, 16, 20, 16, 16, 12, 8, 4, 4, 1, 1, 2, 3, 5, 5, 7, 7, 8, 7, 7, 5, 5, 3, 2, 1, 1, 5, 5, 10, 15, 25, 30, 40, 45, 55, 55, 60, 55, 55, 45, 40, 30, 25, 15, 10, 5, 5
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OFFSET

0,5


COMMENTS

The definition of an orbital system is given in A232500. The number of orbitals over n sectors is counted by the swinging factorial A056040.
The break index of an orbital is the sum of the positions of the up steps that are immediately followed by a step which is not an up step. This statistic is an extension of the major index statistic given in A063746 which appears as the even numbered rows here. This reflects the fact that the swinging factorial can be seen as an extension of the central binomial. The break index is different from the major index of the swinging factorial (which is in A274888).


LINKS



EXAMPLE

The length of row n is floor(n^2/4 + 1). Triangle starts:
[n] [k=0,1,2,...] [row sum]
[0] [1] 1
[1] [1] 1
[2] [1, 1] 2
[3] [2, 2, 2] 6
[4] [1, 1, 2, 1, 1] 6
[5] [3, 3, 6, 6, 6, 3, 3] 30
[6] [1, 1, 2, 3, 3, 3, 3, 2, 1, 1] 20
[7] [4, 4, 8, 12, 16, 16, 20, 16, 16, 12, 8, 4, 4] 140
[8] [1, 1, 2, 3, 5, 5, 7, 7, 8, 7, 7, 5, 5, 3, 2, 1, 1] 70
[9] [5, 5, 10, 15, 25, 30, 40, 45, 55, 55, 60, 55, 55, 45, 40, 30,25,15,10,5,5] 630
T(5, 5) = 3 because the three orbitals [1, 1, 1, 1, 0], [1, 1, 0, 1, 1] and [1, 0, 1, 1, 1] have at position 1 and position 4 an upstep which is immediately followed by a step which is not an upstep.


PROG

(Sage) # uses[unit_orbitals from A274709]
# Brute force counting
def orbital_break_index(n):
S = [0]*(n^2//4 + 1)
for u in unit_orbitals(n):
L = [i+1 if u[i] == 1 and u[i+1] != 1 else 0 for i in (0..n2)]
# i+1 because u is 0based
S[sum(L)] += 1
return S
for n in (0..9): print(orbital_break_index(n))


CROSSREFS



KEYWORD

nonn,tabf


AUTHOR



STATUS

approved



