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A261137 Number of set partitions B'_t(n) of {1,2,...,t} into at most n parts, so that no part contains both 1 and t, or both i and i+1 with 1 <= i < t; triangle B'_t(n), t>=0, 0<=n<=t, read by rows. 5
1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 3, 4, 0, 0, 0, 5, 10, 11, 0, 0, 1, 11, 31, 40, 41, 0, 0, 0, 21, 91, 147, 161, 162, 0, 0, 1, 43, 274, 568, 694, 714, 715, 0, 0, 0, 85, 820, 2227, 3151, 3397, 3424, 3425, 0, 0, 1, 171, 2461, 8824, 14851, 17251, 17686, 17721, 17722 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,14

COMMENTS

B'_t(n) is the number of sequences of t non-identity top-to-random shuffles that leave a deck of n cards invariant.

B'_t(n) = <chi^t, 1_{Sym_n}> where chi is the degree n-1 constituent of the natural permutation character of the symmetric group Sym_n. This gives a combinatorial interpretation of B'_t(n) using sequences of box moves on Young diagrams.

B'_t(t) is the number of set partitions of a set of size t into parts of size at least 2 (A000296); this is also the number of cyclically spaced partitions of a set of size t.

B'_t(n) = B'_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.

D. E. Knuth and O. P. Lossers, Partitions of a circular set, Problem 11151 in Amer. Math. Monthly 114 (3), (2007), p 265, E_4.

FORMULA

B'_t(n) = Sum_{i=0..n} A261139(t,i).

EXAMPLE

Triangle starts:

  1;

  0, 0;

  0, 0, 1;

  0, 0, 0,  1;

  0, 0, 1,  3,   4;

  0, 0, 0,  5,  10,   11;

  0, 0, 1, 11,  31,   40,   41;

  0, 0, 0, 21,  91,  147,  161,  162;

  0, 0, 1, 43, 274,  568,  694,  714,  715;

  0, 0, 0, 85, 820, 2227, 3151, 3397, 3424, 3425;

  ...

MAPLE

g:= proc(t, l, h) option remember; `if`(t=0, `if`(l=1, 0, x^h),

       add(`if`(j=l, 0, g(t-1, j, max(h, j))), j=1..h+1))

    end:

B:= t-> (p-> seq(add(coeff(p, x, j), j=0..i), i=0..t))(g(t, 0$2)):

seq(B(t), t=0..12);  # Alois P. Heinz, Aug 10 2015

MATHEMATICA

StirPrimedGF[0, x_] := 1; StirPrimedGF[1, x_] := 0;

StirPrimedGF[n_, x_] := x^n/(1 + x)*Product[1/(1 - j*x), {j, 1, n - 1}];

StirPrimed[0, 0] := 1; StirPrimed[0, _] := 0;

StirPrimed[t_, n_] := Coefficient[Series[StirPrimedGF[n, x], {x, 0, t}], x^t];

BPrimed[t_, n_] := Sum[StirPrimed[t, m], {m, 0, n}]

(* Second program: *)

g[t_, l_, h_] := g[t, l, h] = If[t == 0, If[l == 1, 0, x^h], Sum[If[j == l, 0, g[t - 1, j, Max[h, j]]], {j, 1, h + 1}]];

B[t_] := Function[p, Table[Sum[Coefficient[p, x, j], {j, 0, i}], {i, 0, t}] ][g[t, 0, 0]];

Table[B[t], {t, 0, 12}] // Flatten (* Jean-Fran├žois Alcover, May 20 2016, after Alois P. Heinz *)

CROSSREFS

Cf. A000296, A261139.

For columns n=3-8 see: A001045, A006342, A214142, A214167, A214188, A214239.

Sequence in context: A238797 A025120 A025096 * A319341 A086798 A155061

Adjacent sequences:  A261134 A261135 A261136 * A261138 A261139 A261140

KEYWORD

nonn,tabl

AUTHOR

Mark Wildon, Aug 10 2015

STATUS

approved

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Last modified August 15 15:31 EDT 2022. Contains 356148 sequences. (Running on oeis4.)