A261318 - OEIS

%I #23 Mar 11 2019 11:40:46

%S 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,

%T 0,273,1491,1365,371,35,1,0,1,820,7728,11046,4704,756,48,1,0,0,2460,

%U 39460,85050,53382,13020,1380,63,1

%N Number of set partitions T'_t(n) of {1,2,...,t} into exactly n parts, with an even number of elements in each part distinguished by marks, and such that no part contains both 1 and t with 1 unmarked or both i and i+1 with i+1 unmarked for some i with 1 <= i < t; triangle T'_t(n), t>=0, 0<=n<=t, read by rows.

%C 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.

%H John R. Britnell and Mark Wildon, <a href="http://arxiv.org/abs/1507.04803">Bell numbers, partition moves and the eigenvalues of the random-to-top shuffle in Dynkin Types A, B and D</a>, arXiv:1507.04803 [math.CO], 2015.

%F 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.

%F G.f. for column n>1: x^n/((1+x)*Product_{j=1..n-1} 1/(1-(2*j-1)*x)).

%F Asymptotically for n > 1: T'_t(n) equals (2n-1)^t/2^n n!

%e Triangle starts:

%e 1;

%e 0,  0;

%e 0,  1,   1;

%e 0,  0,   3,    1;

%e 0,  1,  10,    8,     1;

%e 0,  0,  30,   50,    15,    1;

%e 0,  1,  91,  280,   155,   24,   1;

%e 0,  0, 273, 1491,  1365,  371,  35,  1;

%e 0,  1, 820, 7728, 11046, 4704, 756, 48,  1;

%t 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 T[0, 0] := 1; T[_, 0] := 0; T[0,_]:=0; T[t_, n_] := Coefficient[Series[TGF[n, x], {x, 0, t}], x^t]

%o (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));}

%o tabl(nn) = {for (t=0, nn, for (n=0, t, print1(T(t, n), ", ");); print(););} \\ _Michel Marcus_, Aug 17 2015

%Y Cf. A261275, A261139, A261137.

%K tabl,nonn

%O 0,9

%A _Mark Wildon_, Aug 14 2015

%E One more row by _Michel Marcus_, Aug 17 2015

%E Corrected description in name to agree with section 4.1 in linked paper _Mark Wildon_, Mar 11 2019