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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. 0
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 (list; table; graph; refs; listen; history; text; internal format)
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 A262174

Adjacent sequences:  A261316 A261317 A261318 * A261320 A261321 A261322

KEYWORD

nonn,tabl

AUTHOR

Mark Wildon, Aug 14 2015

STATUS

approved

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Last modified March 29 14:48 EDT 2020. Contains 333107 sequences. (Running on oeis4.)