cp's OEIS Frontend

A274708 A statistic on orbital systems over n sectors: the number of orbitals with k peaks.

%I A274708 #27 Mar 27 2020 06:58:01
%S A274708 1,1,2,4,2,4,2,12,15,3,10,8,2,38,68,30,4,26,30,12,2,121,272,183,49,5,
%T A274708 70,104,60,16,2,384,1026,912,372,72,6,192,350,260,100,20,2,1214,3727,
%U A274708 4095,2220,650,99,7,534,1152,1050,520,150,24,2,3822,13200,17178,11600,4510,1032,130,8
%N A274708 A statistic on orbital systems over n sectors: the number of orbitals with k peaks.
%C A274708 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 A274708 An orbital w has a 'peak' at i+1 when signum(w[i]) < signum(w[i+1]) and signum(w[i+1]) > signum(w[i+2]).
%C A274708 A097692 is a subtriangle.
%H A274708 Peter Luschny, <a href="https://oeis.org/wiki/User:Peter_Luschny/Orbitals">Orbitals</a>
%e A274708 Triangle read by rows, n>=0. The length of row n is floor((n+1)/2) for n>=1.
%e A274708 [ n] [k=0,1,2,...]               [row sum]
%e A274708 [ 0] [  1]                           1
%e A274708 [ 1] [  1]                           1
%e A274708 [ 2] [  2]                           2
%e A274708 [ 3] [  4,    2]                     6
%e A274708 [ 4] [  4,    2]                     6
%e A274708 [ 5] [ 12,   15,   3]               30
%e A274708 [ 6] [ 10,    8,   2]               20
%e A274708 [ 7] [ 38,   68,  30,   4]         140
%e A274708 [ 8] [ 26,   30,  12,   2]          70
%e A274708 [ 9] [121,  272, 183,  49,  5]     630
%e A274708 [10] [ 70,  104,  60,  16,  2]     252
%e A274708 [11] [384, 1026, 912, 372, 72, 6] 2772
%e A274708 [12] [192,  350, 260, 100, 20, 2]  924
%e A274708 T(6, 2) = 2 because the two orbitals [-1, 1, -1, 1, -1, 1] and [1, -1, 1, -1, 1, -1] have 2 peaks.
%o A274708 (Sage) # uses[unit_orbitals from A274709]
%o A274708 # Brute force counting
%o A274708 def orbital_peaks(n):
%o A274708     if n == 0: return [1]
%o A274708     S = [0]*((n+1)//2)
%o A274708     for u in unit_orbitals(n):
%o A274708         L = [1 if sgn(u[i]) < sgn(u[i+1]) and sgn(u[i+1]) > sgn(u[i+2]) else 0 for i in (0..n-3)]
%o A274708         S[sum(L)] += 1
%o A274708     return S
%o A274708 for n in (0..12): print(orbital_peaks(n))
%Y A274708 Cf. A025565 (even col. 0), A056040 (row sum), A097692, A232500.
%Y A274708 Other orbital statistics: A241477 (first zero crossing), A274706 (absolute integral), A274709 (max. height), A274710 (number of turns), A274878 (span), A274879 (returns), A274880 (restarts), A274881 (ascent).
%K A274708 nonn,tabf
%O A274708 0,3
%A A274708 _Peter Luschny_, Jul 10 2016