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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) = 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-François 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.)