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A207978
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Number of n X 2 nonnegative integer arrays with new values 0 upwards introduced in row major order and no element equal to any diagonal or antidiagonal neighbor (colorings ignoring permutations of colors).
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2
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1, 2, 7, 67, 1080, 25287, 794545, 31858034, 1573857867, 93345011951, 6514819011216, 526593974392123, 48658721593531669, 5084549201524804642, 595348294459678745663, 77500341343460209843627, 11140107960738185817545800, 1757660562895916320583653791
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OFFSET
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0,2
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COMMENTS
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a(n) is equal to the number of set partitions of {1,2,...,2n} such that k and k+2 do not appear in the same block for any k. - Andrew Howroyd , May 23 2023
a(n) is equal to the number of set partitions of {1,2,...,2n} such that the only sets of size 1 in the set partition are either {1} or {2}.
a(n) is also the dimension of the centralizer algebra End_{S_m}((V^{(m-1,1)}_{S_m})^{\otimes n-1} \otimes V_m ) where V^{(m-1,1)}_{S_m} is an irreducible S_m module indexed by (m-1,1) and V_m is the permutation module for S_m (with the condition that m is sufficiently large). - Mike Zabrocki, May 23 2023
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LINKS
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FORMULA
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a(n) = Sum_{s=0..2n} (-1)^s binomial(2n-2,s) Bell(2n-s). - Mike Zabrocki, May 23 2023
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EXAMPLE
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Some solutions for n=4:
..0..0....0..0....0..0....0..0....0..0....0..0....0..1....0..0....0..0....0..1
..1..1....1..1....1..1....1..1....1..1....1..1....0..1....1..1....1..1....2..1
..2..3....2..0....2..0....2..2....0..2....0..0....0..1....0..2....0..2....0..1
..2..0....3..3....1..0....3..4....3..1....1..1....0..1....3..4....0..1....0..1
The set partitions of 4 where at most {1} and {2} are the only sets of size 1 are {1234}, {1|234}, {2|134}, {12|34}, {13|24}, {14|23}, {1|2|34} - Mike Zabrocki, May 23 2023
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MAPLE
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a:=n->add((-1)^s*binomial(2*n-2, s) * combinat[bell](2*n-s), s = 0 .. 2*n); # Mike Zabrocki, May 23 2023
# second Maple program:
b:= proc(n) option remember; `if`(n=0, 1,
add(b(n-j)*binomial(n-1, j-1), j=1..n))
end:
a:= n-> `if`(n=0, 1, b(2*n-1)+b(2*n-2)):
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MATHEMATICA
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b[n_] := b[n] = If[n == 0, 1, Sum[b[n-j] Binomial[n-1, j-1], {j, 1, n}]];
a[n_] := If[n == 0, 1, b[2n-1] + b[2n-2]];
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PROG
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(Sage) a = lambda n: sum((-1)**s*binomial(2*n-2, s)*bell_number(2*n-s) for s in range(2*n-2+1)) # Mike Zabrocki, May 23 2023
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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New description and a formula added by Mike Zabrocki, May 23 2023
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STATUS
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approved
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