cp's OEIS Frontend

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

%I A274881 #18 Mar 23 2020 12:09:58
%S A274881 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,
%T A274881 150,30,0,6,120,90,30,6,0,126,1470,882,252,42,0,7,350,371,147,42,7,0,
%U A274881 196,5292,4508,1568,392,56,0,8,1008,1456,672,224,56,8
%N A274881 A statistic on orbital systems over n sectors: the number of orbitals which have an ascent of length k.
%C A274881 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.
%C A274881 The ascent of an orbital is its longest up-run.
%H A274881 Peter Luschny, <a href="https://oeis.org/wiki/User:Peter_Luschny/Orbitals">Orbitals</a>
%e A274881 Triangle read by rows, n>=0. The length of row n is floor((n+2)/2).
%e A274881 [ n] [k=0,1,2,...]                [row sum]
%e A274881 [ 0] [1]                              1
%e A274881 [ 1] [1]                              1
%e A274881 [ 2] [0, 2]                           2
%e A274881 [ 3] [0, 6]                           6
%e A274881 [ 4] [0, 3, 3]                        6
%e A274881 [ 5] [0, 18, 12]                     30
%e A274881 [ 6] [0, 4, 12, 4]                   20
%e A274881 [ 7] [0, 40, 80, 20]                140
%e A274881 [ 8] [0, 5, 40, 20, 5]               70
%e A274881 [ 9] [0, 75, 375, 150, 30]          630
%e A274881 [10] [0, 6, 120, 90, 30, 6]         252
%e A274881 [11] [0, 126, 1470, 882, 252, 42]  2772
%e A274881 [12] [0, 7, 350, 371, 147, 42, 7]   924
%e A274881 T(6,3) = 4 because four orbitals over six sectors have a maximal up-run of length 3.
%e A274881 [-1,-1,-1,1,1,1], [-1,-1,1,1,1,-1], [-1,1,1,1,-1,-1], [1,1,1,-1,-1,-1].
%o A274881 (Sage) # uses[unit_orbitals from A274709]
%o A274881 # Brute force counting
%o A274881 def orbital_ascent(n):
%o A274881     if n < 2: return [1]
%o A274881     S = [0]*((n+2)//2)
%o A274881     for u in unit_orbitals(n):
%o A274881         B = [0]*n
%o A274881         for i in (0..n-1):
%o A274881             B[i] = 0 if u[i] <= 0 else B[i-1] + u[i]
%o A274881         S[max(B)] += 1
%o A274881     return S
%o A274881 for n in (0..12): print(orbital_ascent(n))
%Y A274881 Cf. A056040 (row sum), A232500.
%Y A274881 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).
%K A274881 nonn,tabf
%O A274881 0,4
%A A274881 _Peter Luschny_, Jul 12 2016