A321512 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.

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

Offset

1

Comments

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.

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Index entries for characteristic functions

Example

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.

Prog

(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

Crossrefs

Cf. A024222, A123320, A049206, A321580.

Sequence in context: A354108 A181101 A379274 * A373489 A297054 A359349

Adjacent sequences: A321509 A321510 A321511 * A321513 A321514 A321515

Keyword

nonn

Author

Pedro Menezes, Nov 11 2018

Status

approved