OFFSET
0,9
COMMENTS
T'_t(n) is the number of sequences of t non-identity 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 and every card must be moved at least once.
LINKS
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
T'_t(n) = 1/2^n n! sum(k=0..n-1,binomial(n,k)*(-1)^k*(2(n-k)-1)^t)+(-1)^(n+t)/2^n! for n > 1.
G.f. for column n>1: x^n/((1+x)*Product_{j=1..n-1} 1/(1-(2*j-1)*x)).
Asymptotically for n > 1: T'_t(n) equals (2n-1)^t/2^n n!
EXAMPLE
Triangle starts:
1;
0, 0;
0, 1, 1;
0, 0, 3, 1;
0, 1, 10, 8, 1;
0, 0, 30, 50, 15, 1;
0, 1, 91, 280, 155, 24, 1;
0, 0, 273, 1491, 1365, 371, 35, 1;
0, 1, 820, 7728, 11046, 4704, 756, 48, 1;
MATHEMATICA
TGF[1, x_] := x^2/(1 - x^2); TGF[n_, x_] := x^n/(1 + x)*Product[1/(1 - (2*j - 1)*x), {j, 1, n}];
T[0, 0] := 1; T[_, 0] := 0; T[0, _]:=0; T[t_, n_] := Coefficient[Series[TGF[n, x], {x, 0, t}], x^t]
PROG
(PARI) T(t, n) = {if ((t==0) && (n==0), return(1)); if (n==0, return(0)); if (n==1, return(1 - t%2)); 1/(2^n*n!)*(sum(k=0, n-1, binomial(n, k)*(-1)^k*(2*(n-k)-1)^t)+(-1)^(n+t)); }
tabl(nn) = {for (t=0, nn, for (n=0, t, print1(T(t, n), ", "); ); print(); ); } \\ Michel Marcus, Aug 17 2015
CROSSREFS
KEYWORD
tabl,nonn
AUTHOR
Mark Wildon, Aug 14 2015
EXTENSIONS
One more row by Michel Marcus, Aug 17 2015
Corrected description in name to agree with section 4.1 in linked paper Mark Wildon, Mar 11 2019
STATUS
approved