OFFSET
0,5
COMMENTS
C_t(n) is the number of sequences of t top-to-random shuffles that leave a deck of n cards invariant, if each shuffle is permitted to flip the orientation of the card it moves.
C_t(n) = <pi^t, 1_{BSym_n}> where pi is the permutation character of the hyperoctahedral group BSym_n = C_2 wreath Sym_n given by its imprimitive action on a set of size 2n. This gives a combinatorial interpretation of C_t(n) using sequences of box moves on pairs of Young diagrams.
C_t(t) is the number of set partitions of a set of size t with an even number of elements in each part distinguished by marks.
C_t(n) = C_t(t) if n > t.
LINKS
Alois P. Heinz, Rows n = 0..140, flattened
John R. Britnell and Mark Wildon, Bell numbers, partition moves and the eigenvalues of the random-to-top shuffle in Dynkin Types A, B and D, arXiv:1507.04803 [math.CO], 2015.
FORMULA
G.f.: sum(t>=0, n>=0, C_t(n)x^t/t!y^n) = exp(y/2 (exp(2*x)-1))/(1-y).
C_t(n) = Sum_{i=0..n} A075497(t,i).
EXAMPLE
Triangle starts:
1;
0, 1;
0, 2, 3;
0, 4, 10, 11;
0, 8, 36, 48, 49;
0, 16, 136, 236, 256, 257;
0, 32, 528, 1248, 1508, 1538, 1539;
0, 64, 2080, 6896, 9696, 10256, 10298, 10299;
...
MAPLE
with(combinat):
b:= proc(n, i) option remember; expand(`if`(n=0, 1,
`if`(i<1, 0, add(x^j*multinomial(n, n-i*j, i$j)/j!*add(
binomial(i, 2*k), k=0..i/2)^j*b(n-i*j, i-1), j=0..n/i))))
end:
T:= n-> (p-> seq(add(coeff(p, x, j), j=0..i), i=0..n))(b(n$2)):
seq(T(n), n=0..12); # Alois P. Heinz, Aug 13 2015
MATHEMATICA
CC[t_, n_] := Sum[2^(t - m)*StirlingS2[t, m], {m, 0, n}];
Table[CC[t, n], {t, 0, 12}, {n, 0, t}] // Flatten
(* Second program: *)
multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[x^j*multinomial[n, Join[{n - i*j}, Table[i, j]]]/j!*Sum[Binomial[i, 2*k], {k, 0, i/2}]^j*b[n - i*j, i - 1], {j, 0, n/i}]]];
T[n_] := Function[p, Table[Sum[Coefficient[p, x, j], {j, 0, i}], {i, 0, n} ] ][b[n, n]];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Nov 07 2017, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Mark Wildon, Aug 13 2015
STATUS
approved