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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
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.
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