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A321512
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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.
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2
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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, 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, 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
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OFFSET
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1
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COMMENTS
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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.
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LINKS
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EXAMPLE
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For example, for n = 4, we have the following sequence of shuffles:
c(1) = 1234 <- initial order of cards
c(2) = 2413
c(3) = 4321 <- here's the reverse of c(1)
c(4) = 3142
c(5) = 1234
Hence the characteristic function at n = 4 is 1.
For n = 5,
c(1) = 12345
c(2) = 24135
c(3) = 43215
c(4) = 31425
c(5) = 12345
Observe that for n = 5, there's no 54321 in the c(i) sequence, so the characteristic function at n = 5 is 0.
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PROG
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(Python)
for n in range(1, 101):
cards = [i for i in range(1, n + 1)]
reverse = cards[::-1]
shuffled = cards.copy()
reversein = False
for i in range(n):
evens = shuffled[1::2]
odds = shuffled[0::2]
shuffled = evens + odds
if shuffled == reverse:
reversein = True
print(n, int(reversein))
(PARI)
shuffle(v) = {my(h=#v\2); vector(#v, i, if(i<=h, 2*i, 2*(i-h)-1))};
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)};
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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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