A261319 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 and such that no part contains both 1 and t (each unmarked) or both i and i+1 (each unmarked) for some i with 1 <= i < t; triangle C'_t(n), t>=0, 0<=n<=t, read by rows.

1, 0, 0, 0, 1, 2, 0, 0, 3, 4, 0, 1, 11, 19, 20, 0, 0, 30, 80, 95, 96, 0, 1, 92, 372, 527, 551, 552, 0, 0, 273, 1764, 3129, 3500, 3535, 3536, 0, 1, 821, 8549, 19595, 24299, 25055, 25103, 25104

Offset

0,6

Comments

C'_t(n) is the number of sequences of t non-identity 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-1_{BSym_n})^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 and such that no part contains both 1 and t (each unmarked) or both i and i+1 (each unmarked) for some i with 1 <= i < t.

C'_t(n) = C'_t(t) if n > t.

Links

Table of n, a(n) for n=0..44.

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

C'_t(n) + C'_t(n-1) = Sum_{s=0..t-1} binomial(t-1,s)*A261275(s,n-1) for n>=1.

E.g.f.: diagonal is exp(1/2*(exp(2*x)-2*x-1)).

C'_t(n) = Sum_{i=0..n} A261318(t,i).

Example

Triangle starts:
1;
0,  0;
0,  1,   2;
0,  0,   3,    4;
0,  1,  11,   19,    20;
0,  0,  30,   80,    95,    96;
0,  1,  92,  372,   527,   551,   552;
0,  0, 273, 1764,  3129,  3500,  3535,  3536;
0,  1, 821, 8549, 19595, 24299, 25055, 25103, 25104;

Mathematica

TGF[1, x_] := x^2/(1 - x^2); TGF[n_, x_] := x^n/(1 + x)*Product[1/(1 - (2*j - 1)*x), {j, 1, n}];
T[0, 0] := 1; T[_, 0] := 0; T[0, _] := 0; T[t_, n_] := Coefficient[Series[TGF[n, x], {x, 0, t}], x^t];
CC[t_, n_] := Sum[T[t, m], {m, 0, n}]

Crossrefs

Cf. A261275, A261318, A261139, A261137.

Sequence in context: A188429 A188430 A013585 * A230414 A053653 A332629

Adjacent sequences: A261316 A261317 A261318 * A261320 A261321 A261322

Keyword

nonn,tabl

Author

Mark Wildon, Aug 14 2015

Status

approved