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Search: id:A007238
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| A007238 |
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Length of longest chain of subgroups in S_n.
(Formerly M0945)
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+0
1
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| 0, 1, 2, 4, 5, 6, 7, 10, 11, 12, 13, 15, 16, 17, 18, 22, 23, 24, 25, 27, 28, 29, 30, 33, 34, 35, 36, 38, 39, 40, 41, 46, 47, 48, 49, 51, 52, 53, 54, 57, 58, 59, 60, 62, 63, 64, 65, 69, 70, 71, 72, 74, 75, 76, 77, 80, 81, 82, 83, 85, 86, 87, 88, 94, 95, 96, 97, 99, 100, 101
(list; graph; listen)
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OFFSET
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1,3
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COMMENT
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Starting at a(2), this is column 2 of Table 1 of the Donald M. Davis paper, p.32; the other columns show that the sequence is nearly understood in a deeper sense. The numbers e_p(k,n) defined as min{nu_p(S(k,j)j!): j >= n} appear frequently in algebraic topology. Here S(k,j) is the Stirling number of the second kind and nu_p(-) the exponent of p. The author and Sun proved that if L is sufficiently large, then e_p((p-1)p^L + n -1, n) >= n-1+nu_p([n/p]!). In this paper, we determine the set of integers n for which equality holds in this inequality when p=2 and 3. The condition is roughly that, in the base-p expansion of n, the sum of two consecutive digits must always be less than p. - Jonathan Vos Post (jvospost3(AT)gmail.com), Jul 17 2008
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REFERENCES
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J.-P. Allouche and J. Shallit, The ring of k-regular sequences, II, Theoret. Computer Sci., 307 (2003), 3-29.
L. Babai, On the length of subgroup chains in the symmetric group, Commun. Algebra, 14 (1986), 1729-1736.
Peter J. Cameron; Ron Solomon; Alexandre Turull, Chains of subgroups in symmetric groups, J. Algebra 127 (1989), no. 2, 340-352.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..1000
J.-P. Allouche and J. Shallit, The Ring of k-regular Sequences, II
Donald M. Davis, Divisibility by 2 and 3 of certain Stirling numbers, Jul 16, 2008.
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FORMULA
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ceiling(3n/2) - b(n) - 1, where b(n) = # 1's in binary expansion of n (A000120).
G.f.: 1/(1-x) * (-1/(1-x^2) + Sum(k>=0, x^2^k/(1-x^2^k))). - Ralf Stephan (ralf(AT)ark.in-berlin.de), Apr 13 2002
a(n) = ceiling(3n/2) - b(n) - 1, where b(n) = # 1's in binary expansion of n (A000120). - Jonathan Vos Post (jvospost3(AT)gmail.com), Jul 17 2008
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MAPLE
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A000120 := proc(n) local w, m, i: w := 0: m := n: while m > 0 do i := m mod 2: w := w+i: m := (m-i)/2: od: w: end: for n from 1 to 100 do printf(`%d, `, ceil(3*n/2) - A000120(n) - 1) od:
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CROSSREFS
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Sequence in context: A126424 A138620 A098166 this_sequence A083875 A165291 A080762
Adjacent sequences: A007235 A007236 A007237 this_sequence A007239 A007240 A007241
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KEYWORD
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nonn,nice
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com), Simon Plouffe (simon.plouffe(AT)gmail.com)
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EXTENSIONS
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More terms from James A. Sellers (sellersj(AT)math.psu.edu), Feb 19 2001
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