A261318 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.

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, 0, 0, 2460, 39460, 85050, 53382, 13020, 1380, 63, 1

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

Table of n, a(n) for n=0..54.

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

Cf. A261275, A261139, A261137.

Sequence in context: A294219 A091480 A034374 * A103879 A322706 A051722

Adjacent sequences: A261315 A261316 A261317 * A261319 A261320 A261321

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