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A261139 S'_t(n) is the number of set partitions of {1,2,...,t} into exactly n parts such that no part contains both 1 and t or both i and i+1 for some i with 1 <= i < t; triangle S'_t(n), t >= 0, 0 <= n <= t, read by rows. 10
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, 0, 0, 1, 170, 2290, 6363, 6027, 2400, 435, 35, 1 (list; table; graph; refs; listen; history; text; internal format)
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
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
Sum_{n=0..t} (n-1)*S'_{t-1}(n) + n*S'_{t-2}(n) = A000296(t) for t >= 3. - Yuchun Ji, Feb 23 2021
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
Columns n=3,4 give: A000975, A243869.
Row sums give A000296.
Cf. A261137.
The same as A105794, except for the first two columns.
Sequence in context: A350488 A212868 A184616 * A065860 A363494 A010110
KEYWORD
nonn,tabl
AUTHOR
Mark Wildon, Aug 10 2015
STATUS
approved

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Last modified March 28 09:04 EDT 2024. Contains 371240 sequences. (Running on oeis4.)