|
%I #14 Mar 27 2020 13:58:48
%S 1,0,1,0,2,0,6,0,4,2,0,20,10,0,10,8,2,0,70,56,14,0,28,28,12,2,0,252,
%T 252,108,18,0,84,96,54,16,2,0,924,1056,594,176,22,0,264,330,220,88,20,
%U 2,0,3432,4290,2860,1144,260,26,0,858,1144,858,416,130,24,2
%N Triangle read by rows: number of orbitals over n sectors which have a Catalan decomposition into k parts.
%C The definition of an orbital system is given in A232500.
%C The Catalan decomposition of an orbital w is a list of orbitals which are alternately entirely above or below the main circle ('above' and 'below' in the weak sense) such that their concatenation equals w. If a zero is on the border of two orbitals then it is allocated to the first one. By convention T(0,0) = 1.
%C The number of orbitals over n sectors is counted by the swinging factorial A056040.
%H Peter Luschny, <a href="https://oeis.org/wiki/User:Peter_Luschny/Orbitals">Orbitals</a>
%F T(n,1) = 2*floor((n+2)/2)*n!/floor((n+2)/2)!^2 = A241543(n+2) for n>=2.
%F For odd n>1 T(n,1) = Sum_{k>=0} T(n+1,k).
%F A056040(n) - T(n,1) = A232500(n) for n>=2.
%F Main diagonal: T(n, floor(n/2)) = A266722(n) for n>1.
%F A275326(n,k) = ceiling(T(n,k)/2).
%e Table starts:
%e [ n] [k=0,1,2,...] [row sum]
%e [ 0] [1] 1
%e [ 1] [0, 1] 1
%e [ 2] [0, 2] 2
%e [ 3] [0, 6] 6
%e [ 4] [0, 4, 2] 6
%e [ 5] [0, 20, 10] 30
%e [ 6] [0, 10, 8, 2] 20
%e [ 7] [0, 70, 56, 14] 140
%e [ 8] [0, 28, 28, 12, 2] 70
%e [ 9] [0, 252, 252, 108, 18] 630
%e [10] [0, 84, 96, 54, 16, 2] 252
%e [11] [0, 924, 1056, 594, 176, 22] 2772
%e [12] [0, 264, 330, 220, 88, 20, 2] 924
%e For example T(2*n, n) = 2 counts the Catalan decompositions
%e [[-1, 1], [1, -1], [-1, 1], ..., [(-1)^n, (-1)^(n+1)]] and
%e [[1, -1], [-1, 1], [1, -1], ..., [(-1)^(n+1), (-1)^n]].
%o (Sage) # uses[unit_orbitals from A274709]
%o # Brute force counting
%o from itertools import accumulate
%o def catalan_factors(P):
%o def bisect(orb):
%o i = 1
%o A = list(accumulate(orb))
%o if orb[1] > 0 if orb[0] == 0 else orb[0] > 0:
%o while i < len(A) and A[i] >= 0: i += 1
%o else:
%o while i < len(A) and A[i] <= 0: i += 1
%o return i
%o R = []
%o while P:
%o i = bisect(P)
%o R.append(P[:i])
%o P = P[i:]
%o return R
%o def orbital_factors(n):
%o if n == 0: return [1]
%o if n == 1: return [0, 1]
%o S = [0]*(n//2 + 1)
%o for o in unit_orbitals(n):
%o S[len(catalan_factors(o))] += 1
%o return S
%o for n in (0..9): print(orbital_factors(n))
%Y Cf. A056040 (row sum), A241543, A232500, A266722, A275326.
%Y 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 nonn,tabf
%O 0,5
%A _Peter Luschny_, Aug 15 2016
|
