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A261137
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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.
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5
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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
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OFFSET
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0,14
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COMMENTS
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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.
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LINKS
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FORMULA
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B'_t(n) = Sum_{i=0..n} A261139(t,i).
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EXAMPLE
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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;
...
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MAPLE
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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)):
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MATHEMATICA
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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]];
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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