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

%I A275333 #24 Mar 27 2020 12:11:21
%S A275333 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,
%T A275333 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,
%U A275333 25,30,40,45,55,55,60,55,55,45,40,30,25,15,10,5,5
%N A275333 Triangle read by rows, the break statistic on orbital systems over n sectors.
%C A275333 The definition of an orbital system is given in A232500. The number of orbitals over n sectors is counted by the swinging factorial A056040.
%C A275333 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).
%H A275333 Peter Luschny, <a href="https://oeis.org/wiki/User:Peter_Luschny/Orbitals">Orbitals</a>
%e A275333 The length of row n is floor(n^2/4 + 1). Triangle starts:
%e A275333 [n] [k=0,1,2,...] [row sum]
%e A275333 [0] [1] 1
%e A275333 [1] [1] 1
%e A275333 [2] [1, 1] 2
%e A275333 [3] [2, 2, 2] 6
%e A275333 [4] [1, 1, 2, 1, 1] 6
%e A275333 [5] [3, 3, 6, 6, 6, 3, 3] 30
%e A275333 [6] [1, 1, 2, 3, 3, 3, 3, 2, 1, 1] 20
%e A275333 [7] [4, 4, 8, 12, 16, 16, 20, 16, 16, 12, 8, 4, 4] 140
%e A275333 [8] [1, 1, 2, 3, 5, 5, 7, 7, 8, 7, 7, 5, 5, 3, 2, 1, 1] 70
%e A275333 [9] [5, 5, 10, 15, 25, 30, 40, 45, 55, 55, 60, 55, 55, 45, 40, 30,25,15,10,5,5] 630
%e A275333 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.
%o A275333 (Sage) # uses[unit_orbitals from A274709]
%o A275333 # Brute force counting
%o A275333 def orbital_break_index(n):
%o A275333     S = [0]*(n^2//4 + 1)
%o A275333     for u in unit_orbitals(n):
%o A275333         L = [i+1 if u[i] == 1 and u[i+1] != 1 else 0 for i in (0..n-2)]
%o A275333         #    i+1 because u is 0-based
%o A275333         S[sum(L)] += 1
%o A275333     return S
%o A275333 for n in (0..9): print(orbital_break_index(n))
%Y A275333 Cf. A056040 (row sum), A063746 (sub triangle), A274888 (q-swinging factorial).
%Y A275333 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).
%K A275333 nonn,tabf
%O A275333 0,5
%A A275333 _Peter Luschny_, Jul 23 2016