A275333 Triangle read by rows, the break statistic on orbital systems over n sectors.

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

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

Table of n, a(n) for n=0..79.

Peter Luschny, Orbitals

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 up-step which is immediately followed by a step which is not an up-step.

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..n-2)]
        #    i+1 because u is 0-based
        S[sum(L)] += 1
    return S
for n in (0..9): print(orbital_break_index(n))

Crossrefs

Cf. A056040 (row sum), A063746 (sub triangle), A274888 (q-swinging factorial).

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), A274881 (ascent).

Sequence in context: A188795 A325444 A058745 * A108393 A327342 A297828

Adjacent sequences: A275330 A275331 A275332 * A275334 A275335 A275336

Keyword

nonn,tabf

Author

Peter Luschny, Jul 23 2016

Status

approved