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A261275 Number of set partitions C_t(n) of {1,2,...,t} into at most n parts, with an even number of elements in each part distinguished by marks; triangle C_t(n), t>=0, 0<=n<=t, read by rows. 4
1, 0, 1, 0, 2, 3, 0, 4, 10, 11, 0, 8, 36, 48, 49, 0, 16, 136, 236, 256, 257, 0, 32, 528, 1248, 1508, 1538, 1539, 0, 64, 2080, 6896, 9696, 10256, 10298, 10299, 0, 128, 8256, 39168, 66384, 74784, 75848, 75904, 75905 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

C_t(n) is the number of sequences of t 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.

C_t(n) = <pi^t, 1_{BSym_n}> where pi is the permutation character of the hyperoctahedral group BSym_n = C_2 wreath Sym_n given by its imprimitive action on a set of size 2n. This gives a combinatorial interpretation of C_t(n) using sequences of box moves on pairs of Young diagrams.

C_t(t) is the number of set partitions of a set of size t with an even number of elements in each part distinguished by marks.

C_t(n) = C_t(t) if n > t.

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.

FORMULA

G.f.: sum(t>=0, n>=0, C_t(n)x^t/t!y^n) = exp(y/2 (exp(2*x)-1))/(1-y).

C_t(n) = Sum_{i=0..n} A075497(t,i).

EXAMPLE

Triangle starts:

1;

0,  1;

0,  2,    3;

0,  4,   10,   11;

0,  8,   36,   48,   49;

0, 16,  136,  236,  256,   257;

0, 32,  528, 1248, 1508,  1538,  1539;

0, 64, 2080, 6896, 9696, 10256, 10298, 10299;

..

MAPLE

with(combinat):

b:= proc(n, i) option remember; expand(`if`(n=0, 1,

       `if`(i<1, 0, add(x^j*multinomial(n, n-i*j, i$j)/j!*add(

        binomial(i, 2*k), k=0..i/2)^j*b(n-i*j, i-1), j=0..n/i))))

    end:

T:= n-> (p-> seq(add(coeff(p, x, j), j=0..i), i=0..n))(b(n$2)):

seq(T(n), n=0..12);  # Alois P. Heinz, Aug 13 2015

MATHEMATICA

CC[t_, n_] := Sum[2^(t - m)*StirlingS2[t, m], {m, 0, n}];

Table[CC[t, n], {t, 0, 12}, {n, 0, t}] // Flatten

(* Second program: *)

multinomial[n_, k_List] := n!/Times @@ (k!);

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[x^j*multinomial[n, Join[{n - i*j}, Table[i, j]]]/j!*Sum[Binomial[i, 2*k], {k, 0, i/2}]^j*b[n - i*j, i - 1], {j, 0, n/i}]]];

T[n_] := Function[p, Table[Sum[Coefficient[p, x, j], {j, 0, i}], {i, 0, n} ] ][b[n, n]];

Table[T[n], {n, 0, 12}] // Flatten (* Jean-Fran├žois Alcover, Nov 07 2017, after Alois P. Heinz *)

CROSSREFS

Columns n=0,1,2,3 give A000007, A000079, A007582, A233162 (proved for n=3 in reference above).

Main diagonal gives A004211.

Cf. A075497.

Sequence in context: A121598 A344276 A258818 * A140326 A261781 A211402

Adjacent sequences:  A261272 A261273 A261274 * A261276 A261277 A261278

KEYWORD

nonn,tabl

AUTHOR

Mark Wildon, Aug 13 2015

STATUS

approved

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Last modified October 23 21:27 EDT 2021. Contains 348217 sequences. (Running on oeis4.)