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A237018
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Number A(n,k) of partitions of the k-dimensional hypercube resulting from a sequence of n bisections, each of which splits any part perpendicular to any of the axes; square array A(n,k), n>=0, k>=0, read by antidiagonals.
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12
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1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 8, 5, 0, 1, 4, 18, 39, 14, 0, 1, 5, 32, 132, 212, 42, 0, 1, 6, 50, 314, 1080, 1232, 132, 0, 1, 7, 72, 615, 3440, 9450, 7492, 429, 0, 1, 8, 98, 1065, 8450, 40320, 86544, 47082, 1430, 0, 1, 9, 128, 1694, 17604, 124250, 494736, 819154, 303336, 4862, 0
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OFFSET
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0,8
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COMMENTS
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The g.f. given below is a generalization of formulas given by Murray R. Bremner and Sara Madariaga in A236339 and A236342. According to them A(n,k) also gives the number of distinct monomials of degree n+1 in the universal algebra with k nonassociative binary products {*1,...,*k} related only by the interchange laws from k-category theory: (a *i b) *j (c *i d) = (a *j c) *i (b *j d) for i,j in {1,...,k} and i<j.
These numbers can be regarded as (one of many possible definitions of) higher-dimensional Catalan numbers. - N. J. A. Sloane, Feb 12 2014
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LINKS
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FORMULA
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G.f. G_k of column k satisfies: (-1)^k*x = Sum_{i=0..k} (-1)^i*C(k,i)*(G_k*x)^(2^(k-i)).
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EXAMPLE
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A(3,1) = 5:
[||-|---], [-|||---], [-|-|-|-], [---|||-], [---|-||].
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A(2,2) = 8:
._______. ._______. ._______. ._______.
| | | | | | | | |_______| | |
| | | | | | | | |_______| |_______|
| | | | | | | | | | |_______|
|_|_|___| |___|_|_| |_______| |_______|
._______. ._______. ._______. ._______.
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|___| | | |___| |___|___| |_______|
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|___|___| |___|___| |_______| |___|___|.
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Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, 1, ...
0, 1, 2, 3, 4, 5, 6, ...
0, 2, 8, 18, 32, 50, 72, ...
0, 5, 39, 132, 314, 615, 1065, ...
0, 14, 212, 1080, 3440, 8450, 17604, ...
0, 42, 1232, 9450, 40320, 124250, 311472, ...
0, 132, 7492, 86544, 494736, 1912900, 5770692, ...
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MAPLE
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A:= (n, k)-> coeff(series(RootOf(x*(-1)^k=add((-1)^i*
binomial(k, i)*(G*x)^(2^(k-i)), i=0..k), G), x, n+1), x, n):
seq(seq(A(n, d-n), n=0..d), d=0..10);
# second Maple program:
b:= proc(n, k, t) option remember; `if`(t=0, 1, `if`(t=1,
A(n-1, k), add(A(j, k)*b(n-j-1, k, t-1), j=0..n-2)))
end:
A:= proc(n, k) option remember; `if`(n=0, 1,
-add(binomial(k, j)*(-1)^j*b(n+1, k, 2^j), j=1..k))
end:
seq(seq(A(n, d-n), n=0..d), d=0..10);
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MATHEMATICA
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b[n_, k_, t_] := b[n, k, t] = If[t == 0, 1, If[t == 1, A[n-1, k], Sum[A[j, k]*b[n-j-1, k, t-1], {j, 0, n-2}]]]; A[n_, k_] := A[n, k] = If[n == 0, 1, -Sum[ Binomial[k, j]*(-1)^j*b[n+1, k, 2^j], {j, 1, k}]]; Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Jan 19 2015, after Alois P. Heinz *)
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CROSSREFS
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Columns k=0-10 give: A000007, A000108, A236339(n+1), A236342(n+1), A237019, A237020, A237021, A237022, A237023, A237024, A237025.
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KEYWORD
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AUTHOR
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STATUS
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approved
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