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
Examples
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;
Links
- 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.
Programs
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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}]
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).
Comments