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%I #24 May 03 2023 15:03:28
%S 1,1,0,1,0,0,0,1,0,1,0,1,0,0,0,1,0,1,0,0,0,0,0,1,0,1,0,1,0,0,0,1,0,0,
%T 0,1,0,0,0,1,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,1,0,1,0,0,0,1,0,1,0,0,
%U 0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,1
%N Characteristic function of the reverse in the shuffle (perfect faro shuffle with cut): 1 if the sequence of shuffles of n cards contains the reverse of the original order of cards, 0 if not.
%C The characteristic function of A321580: 1 if in the sequence of Faro's shuffle of n cards there is at some point the exact reverse of the initial order (the cards are backwards); 0 if not.
%H Antti Karttunen, <a href="/A321512/b321512.txt">Table of n, a(n) for n = 1..65537</a>
%H <a href="/index/Ch#char_fns">Index entries for characteristic functions</a>
%e For example, for n = 4, we have the following sequence of shuffles:
%e c(1) = 1234 <- initial order of cards
%e c(2) = 2413
%e c(3) = 4321 <- here's the reverse of c(1)
%e c(4) = 3142
%e c(5) = 1234
%e Hence the characteristic function at n = 4 is 1.
%e For n = 5,
%e c(1) = 12345
%e c(2) = 24135
%e c(3) = 43215
%e c(4) = 31425
%e c(5) = 12345
%e Observe that for n = 5, there's no 54321 in the c(i) sequence, so the characteristic function at n = 5 is 0.
%o (Python)
%o for n in range(1, 101):
%o cards = [i for i in range(1, n + 1)]
%o reverse = cards[::-1]
%o shuffled = cards.copy()
%o reversein = False
%o for i in range(n):
%o evens = shuffled[1::2]
%o odds = shuffled[0::2]
%o shuffled = evens + odds
%o if shuffled == reverse:
%o reversein = True
%o print(n, int(reversein))
%o (PARI)
%o shuffle(v) = {my(h=#v\2); vector(#v, i, if(i<=h, 2*i, 2*(i-h)-1))};
%o permcycs(v) = {my(f=vector(#v), L=List()); for(i=1, #v, if(!f[i], my(T=List(), j=i); while(!f[j], f[j]=1; listput(T, j); j=v[j]); listput(L, Vec(T)))); Vec(L)};
%o A321512(n)={my(v=permcycs(shuffle([1..n])), e=-1); for(k=1, #v, my(p=v[k]); if(#p>1||n%2==0||2*p[1]<>n+1, my(h=#p\2); if(e<0, e=valuation(#p, 2)); if(valuation(#p, 2)<>e || e==0 || p[1..h]+p[h+1..2*h]<>vector(h, i, n+1), return(0)))); 1}; \\ This is _Andrew Howroyd_'s Nov 13 2018 code for the characteristic function of A321580, given under that entry with the name "ok". Copied here by _Antti Karttunen_, Dec 06 2021
%Y Cf. A024222, A123320, A049206, A321580.
%K nonn
%O 1
%A _Pedro Menezes_, Nov 11 2018
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