

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+1k)*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
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OFFSET

1,4


COMMENTS

a(4k+2) = 0; also, the same sequence enumerates permutations of {0,1,...,n1} 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, AddisonWesley, 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, Rightinvariant metrics applied to rank correlation coefficients, Australas. J. Combin. 57 (2013), 157187; see Section 5 for similar questions about other nonparametric rank correlation coefficients.
E. I. Marshall, Conditions for rank correlation to be zero, Sankhyā 56(1) (1994), 5966; 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 9817, 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 9817, 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(12) (2001), 133145.
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



