The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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
 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 (list; table; graph; refs; listen; history; text; internal format)
 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 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified March 29 14:48 EDT 2020. Contains 333107 sequences. (Running on oeis4.)