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A129847
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b(n) = number of set partitions of {1, 2, ..., n} whose blocks consist only of elements that differ by two or less (that is, have only the forms {i}, {i,i+1}, {i,i+2}, or {i,i+1,i+2}).
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7
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1, 1, 2, 5, 10, 20, 42, 87, 179, 370, 765, 1580, 3264, 6744, 13933, 28785, 59470, 122865, 253838, 524428, 1083466, 2238435, 4624595, 9554390, 19739321, 40781336, 84254032, 174068400, 359624425, 742982225, 1534997482, 3171296957, 6551883314, 13536157460
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OFFSET
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0,3
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COMMENTS
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(1) Bryce Duncan and I found the sequence {b(n)} while studying a graph invariant we call the Bell number. For a simple graph G = (V,E), we define B(G) to be the number of partitions P of V in which each block of P is an independent set of G. The sequence b(n) considered here results from choosing V = {1, 2, ..., n} and E = {(i,j) : |i - j| > 2. (The classical Bell numbers B(n) come from the edgeless graph on n vertices.)
(2) The constant r in the formula is the dominant root of the characteristic equation of a linear homogeneous recurrence relation that also defines {b(n)}. (This recurrence relation, along with initial conditions, appears in the Mathematica program given below. The formula for b(n) is analogous to one version of the Binet formula for the Fibonacci numbers, namely F(n) = the integer nearest to (1/sqrt(5)) p^n where p is the golden mean. The shifted Fibonacci numbers F(n+1) are also graphical Bell numbers: replace the condition |i - j| > 2 with |i - j| > 1 to obtain them.
(3) Bell numbers for simple graphs satisfy the deletion-contraction identity B(G) = B(G\e) - B(G/e), but not the product identity B(G union H) = B(G)B(H) for disjoint graphs G and H. However, we do have B(B join H) = B(G)B(H) for the join of graphs G and H. The join graph G join H results from adding to their disjoint union, all possible edges that join a vertex of G to a vertex of H.
a(n) is the number of compositions (ordered partitions) of n into parts 1, 2, 3, and 4 where there are two kinds of part 3. - Joerg Arndt, Sep 16 2016
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REFERENCES
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Herbert S. Wilf, Generatingfunctiononogy (2nd Edition), Academic Press 1990, pp. 1 - 10 and pp. 20 - 23.
Arthur T. Benjamin and Jennifer J. Quinn, Proofs that Really Count, Dolciani Mathematical Expositions (MAA), (2003), p. 1. (A relevant combinatorial interpretation of Fibonacci numbers.)
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LINKS
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FORMULA
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b(n) = the integer nearest to s r^n, where r = 2.0659948920 ... and s = 0.53979687305... . (More information in comment (2).)
a(n) = Sum_{k=0..floor(n/2)} Sum_{j=0..n-2k} C(n-j-k,k)*C(2k,j). - Paul Barry, Mar 24 2009
G.f.: 1/(1 - x - x^2 - 2*x^3 - x^4). - Colin Barker, May 02 2012
a(n) = a(n-1) + a(n-2) + 2*(n-3) + a(n-4) with a(0) = a(1) = 1, a(2) = 2, a(3) = 5. - Taras Goy, Aug 04 2017
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EXAMPLE
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b(4) = 10, with set partitions {{1},{2},{3}, {4}}; {{1,2}, {3}, {4}}; {{1,3}, {2}, {4}}; {{1}, {2,3}, {4}}; {{1}, {2,4}, {3}}; {{1}, {2}, {3,4}}; {{1,2,3}, {4}}; {{1}, {2,3,4}}; {{1,2}, {3,4}} and {{1,3}, {2,4}}.
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MAPLE
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a:= n-> (<<0|1|0|0>, <0|0|1|0>, <0|0|0|1>, <1|2|1|1>>^n)[4, 4]:
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MATHEMATICA
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b[1] = 1; b[2] = 2; b[3] = 5; b[4] = 10 b[n_] := b[n] = b[n-1] + b[n-2] + 2 b[n-3] + b[n-4] Table[b[n], {n, 1, 30}]
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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Rhodes Peele (rpeele(AT)mail.aum.edu), May 22 2007
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EXTENSIONS
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STATUS
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approved
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