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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.

%I A261275 #25 Jun 18 2022 12:28:08
%S A261275 1,0,1,0,2,3,0,4,10,11,0,8,36,48,49,0,16,136,236,256,257,0,32,528,
%T A261275 1248,1508,1538,1539,0,64,2080,6896,9696,10256,10298,10299,0,128,8256,
%U A261275 39168,66384,74784,75848,75904,75905,0,256,32896,226496,475136,586352,607520,609368,609440,609441
%N 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.
%C A261275 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 A261275 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 A261275 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 A261275 C_t(n) = C_t(t) if n > t.
%H A261275 Alois P. Heinz, <a href="/A261275/b261275.txt">Rows n = 0..140, flattened</a>
%H A261275 John R. Britnell and Mark Wildon, <a href="http://arxiv.org/abs/1507.04803">Bell numbers, partition moves and the eigenvalues of the random-to-top shuffle in Dynkin Types A, B and D</a>, arXiv:1507.04803 [math.CO], 2015.
%F A261275 G.f.: sum(t>=0, n>=0, C_t(n)x^t/t!y^n) = exp(y/2 (exp(2*x)-1))/(1-y).
%F A261275 C_t(n) = Sum_{i=0..n} A075497(t,i).
%e A261275 Triangle starts:
%e A261275   1;
%e A261275   0,  1;
%e A261275   0,  2,    3;
%e A261275   0,  4,   10,   11;
%e A261275   0,  8,   36,   48,   49;
%e A261275   0, 16,  136,  236,  256,   257;
%e A261275   0, 32,  528, 1248, 1508,  1538,  1539;
%e A261275   0, 64, 2080, 6896, 9696, 10256, 10298, 10299;
%e A261275   ...
%p A261275 with(combinat):
%p A261275 b:= proc(n, i) option remember; expand(`if`(n=0, 1,
%p A261275        `if`(i<1, 0, add(x^j*multinomial(n, n-i*j, i$j)/j!*add(
%p A261275         binomial(i, 2*k), k=0..i/2)^j*b(n-i*j, i-1), j=0..n/i))))
%p A261275     end:
%p A261275 T:= n-> (p-> seq(add(coeff(p, x, j), j=0..i), i=0..n))(b(n$2)):
%p A261275 seq(T(n), n=0..12);  # _Alois P. Heinz_, Aug 13 2015
%t A261275 CC[t_, n_] := Sum[2^(t - m)*StirlingS2[t, m], {m, 0, n}];
%t A261275 Table[CC[t, n], {t, 0, 12}, {n, 0, t}] // Flatten
%t A261275 (* Second program: *)
%t A261275 multinomial[n_, k_List] := n!/Times @@ (k!);
%t A261275 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 A261275 T[n_] := Function[p, Table[Sum[Coefficient[p, x, j], {j, 0, i}], {i, 0, n} ] ][b[n, n]];
%t A261275 Table[T[n], {n, 0, 12}] // Flatten (* _Jean-François Alcover_, Nov 07 2017, after _Alois P. Heinz_ *)
%Y A261275 Columns n=0,1,2,3 give A000007, A000079, A007582, A233162 (proved for n=3 in reference above).
%Y A261275 Main diagonal gives A004211.
%Y A261275 Cf. A075497.
%K A261275 nonn,tabl
%O A261275 0,5
%A A261275 _Mark Wildon_, Aug 13 2015