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A056876 Number of permutations (p_1, ..., p_n) of {1,...,n} that are "balanced" in the sense that the sum of k*p_k equals the sum of (n+1-k)*p_k; equivalently, the expected value of k*p_k is (expected value of k) times (expected value of p_k), assuming the uniform distribution. 3
1, 0, 0, 2, 6, 0, 184, 936, 6688, 0, 420480, 4298664, 44405142, 0, 6732621476, 92014579912, 1345077232898, 0, 349174373111790, 6179276762966832, 114913276077265202, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

a(4k+2) = 0; also, the same sequence enumerates permutations of {0,1,...,n-1} with the stated expected value property.

Also, central coefficients in the expansion of the probability generating function for the exact null distribution of Spearman's rho. - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), May 14 2002

REFERENCES

D. E. Knuth, The Art of Computer Programming: Generating all tuples and permutations, Volume 4, Fascicle 2, Addison-Wesley, Upper Saddle River, NJ (2005); p. 74, Exercise 104.

Ivan Moscovich, More Brainmatics Logic Puzzles, see p. 130. - from Neven Juric, Jan 21 2010.

LINKS

Table of n, a(n) for n=1..22.

Dongyang Cheng and Petros Hadjicostas, Right-invariant metrics applied to rank correlation coefficients, Australas. J. Combin. 57 (2013), 157-187; see Section 5 for similar questions about other non-parametric rank correlation coefficients.

E. I. Marshall, Conditions for rank correlation to be zero, Sankhyā 56(1) (1994), 59-66; he proved that a(n) = 0 if and only if n = 3 or n = 4*k+2 for some integer k >= 0 (see Theorem 2, p. 62).

M. A. van de Wiel and A. Di Bucchianico, Fast computation of the exact null distribution of Spearman's rho and Page's L statistic for samples with and without ties, Memorandum COSOR 98-17, 1998, Eindhoven University of Technology.

M. A. van de Wiel and A. Di Bucchianico, Fast computation of the exact null distribution of Spearman's rho and Page's L statistic for samples with and without ties, Memorandum COSOR 98-17, 1998, Eindhoven University of Technology.

M. A. van de Wiel and A. Di Bucchianico, Fast computation of the exact null distribution of Spearman's rho and Page's L statistic for samples with and without ties, J. Statist. Plann. Inference, 92(1-2) (2001), 133-145.

Wikipedia, Spearman rank correlation coefficient.

EXAMPLE

a(5) = 6 because of the permutations 15432, 23451, 25314, 41352, 43215, 51234.

CROSSREFS

Sequence in context: A293016 A122685 A109581 * A255484 A021797 A068959

Adjacent sequences:  A056873 A056874 A056875 * A056877 A056878 A056879

KEYWORD

hard,nonn

AUTHOR

Don Knuth, Sep 03 2000

EXTENSIONS

More terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), May 14 2002

STATUS

approved

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Last modified April 21 14:05 EDT 2021. Contains 343154 sequences. (Running on oeis4.)