OFFSET
0,14
COMMENTS
S'_t(n) is the number of sequences of t non-identity top-to-random shuffles of a deck of n cards that move each card at some time, and overall leave the deck invariant. (See link below.) A261137 may be defined by B'_t(n) = Sum_{m=0..n} S'_t(m).
LINKS
Alois P. Heinz, Rows n = 0..140, flattened
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.
D. E. Knuth and O. P. Lossers, Partitions of a circular set, Problem 11151 in Amer. Math. Monthly 114 (3), (2007), p 265, E_4.
Sophie Morier-Genoud, Counting Coxeter's friezes over a finite field via moduli spaces, arXiv:1907.12790 [math.CO], 2019.
Augustine O. Munagi, Two Applications of the Bijection on Fibonacci Set Partitions, Fibonacci Quart. 55 (2017), no. 5, 144-148. See c(n,k) p. 145 giving shifted triangle.
FORMULA
G.f. for column n > 1: x^n/((1+x)*Product_{j=1..n-1} (1-j*x)).
S'_t(n) ~ (n-1)^t/n! as t tends to infinity.
Recurrence: S'_t(n) = S'_{t-1}(n-1) + (n-1)*S'_{t-1}(n) for n >= 3.
S'_t(n) = (1/n!) * Sum_{j=0..n} (-1)^(n-j) * binomial(n, j) * ((j-1)^t + (-1)^t * (j-1)) for t>0. - Andrew Howroyd, Apr 08 2017
T(m, k) = Sum_{i=k..m} Stirling2(i-1, k-1)*(-1)^(i+m), for k >= 2. (See Peter Bala's original formula at A105794 dated Jul 10 2013.) - Igor Victorovich Statsenko, May 31 2024
T(m, k) = (Sum_{i=0..m} Stirling2(i, k)*binomial(m,i)*(-1)^(m-i))*I(m,k), where I(m,k) = (1-Sum_{i=0..m} Stirling1(k, i))^(m+k) for k >= 0. (See Peter Bala's original formula at A105794 dated Jul 10 2013.) - Igor Victorovich Statsenko, Jun 01 2024
EXAMPLE
Triangle starts:
1;
0, 0;
0, 0, 1;
0, 0, 0, 1;
0, 0, 1, 2, 1;
0, 0, 0, 5, 5, 1;
0, 0, 1, 10, 20, 9, 1;
0, 0, 0, 21, 70, 56, 14, 1;
0, 0, 1, 42, 231, 294, 126, 20, 1;
0, 0, 0, 85, 735, 1407, 924, 246, 27, 1;
...
MAPLE
g:= proc(t, l, h) option remember; `if`(t=0, `if`(l=1, 0, x^h),
add(`if`(j=l, 0, g(t-1, j, max(h, j))), j=1..h+1))
end:
S:= t-> (p-> seq(coeff(p, x, i), i=0..t))(g(t, 0$2)):
seq(S(t), t=0..12); # Alois P. Heinz, Aug 10 2015
MATHEMATICA
StirPrimedGF[n_, x_] := x^n/(1 + x)*Product[1/(1 - j*x), {j, 1, n - 1}]; T[0, 0] = 1; T[_, 0] = T[_, 1] = 0; T[n_, k_] := SeriesCoefficient[ StirPrimedGF[k, x], {x, 0, n}]; Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* script completed by Jean-François Alcover, Jan 31 2016 *)
PROG
(PARI)
a(n, k)=if(k==0, n==0, sum(j=0, k, binomial(k, j) * (-1)^(k-j) * ((j-1)^n + (-1)^n * (j-1))) / k!);
for(n=0, 10, for(k=0, n, print1( a(n, k), ", "); ); print(); ); \\ Andrew Howroyd, Apr 08 2017
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Mark Wildon, Aug 10 2015
STATUS
approved