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 A000140 Kendall-Mann numbers: the most common number of inversions in a permutation on n letters is floor(n*(n-1)/4); a(n) is the number of permutations with this many inversions. (Formerly M1665 N0655) 16
 1, 1, 2, 6, 22, 101, 573, 3836, 29228, 250749, 2409581, 25598186, 296643390, 3727542188, 50626553988, 738680521142, 11501573822788, 190418421447330, 3344822488498265, 62119523114983224, 1214967840930909302, 24965661442811799655, 538134522243713149122 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS Row maxima of A008302, see example. The term a(0) would be 1: the empty product is one and there is just one coefficient 1=x^0, corresponding to the 1 empty permutation (which has 0 inversions). From Ryen Lapham and Anant Godbole, Dec 12 2006: (Start) Also, the number of permutations on {1,2,...,n} for which the number A of monotone increasing subsequences of length 2 and the number D of monotone decreasing 2-subsequences are as close to each other as possible, i.e., 0 or 1. We call such permutations 2-balanced. If 4|n(n-1) then (with A and D as above) the feasible values of A-D are C(n,2), C(n,2)-2,...,2,0,-2,...,-C(n,2), whereas if 4 does not divide n(n-1), A-D may equal C(n,2), C(n,2)-2,...,1,-1,...,-C(n,2). Let a_n(i) equal the number of permutations with A-D the i-th highest feasible value. The sequence in question gives the number of permutations for which A-D=0 or A-D=1, i.e., it equals A_n(j) where j = floor((binomial(n,2)+2)/2). Here is the recursion: a_n(i) = a_n(i-1) + a_{n-1}(i) for 1 <= i <= n and a_n(n+k) = a_n(n+k-1) + a_{n-1}(n+k) - a_n(k) for k >= 1. (End) The only two primes found < 301 are for n = 3 and 6. Define an ordered list to have n terms with terms t(k) for k=1..n. Specify that t(k) ranges from 1 to k, hence the third term t(3) can be 1, 2, or 3. Find all sums of the terms for all n! allowable arrangements to obtain a maximum sum for the greatest number of arrangements. This number is a(n). For n=4, the maximum sum 7 appears in 6 arrangements: 1114, 1123, 1213, 1222, 1231, and 1132. - J. M. Bergot, May 14 2015 Named after the British statistician Maurice George Kendall (1907-1983) and the Austrian-American mathematician Henry Berthold Mann (1905-2000). - Amiram Eldar, Apr 07 2023 REFERENCES F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 241. N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Robert Israel, Robert G. Wilson v and N. J. A. Sloane Table of n, a(n) for n = 1..400 (a(1) to a(61) from Sloane, a(62) to a(350) from Wilson). Dorin Andrica and Ovidiu Bagdasar, On some results concerning the polygonal polynomials, Carpathian Journal of Mathematics (2019) Vol. 35, No. 1, 1-11. Dominique Foata, Distributions eulériennes et mahoniennes sur le groupe des permutations, pp. 27-49 of M. Aigner, editor, Higher Combinatorics, Reidel, Dordrecht, Holland, 1977. Mikhail Gaichenkov, The property of Kendall-Mann numbers, MathOverflow, 2010. Richard K. Guy, Letter to N. J. A. Sloane with attachment, Mar 1988. A. Waksman, On the complexity of inversions, IEEE Trans. Computers, 19 (1970), 1225-1226. Wikipedia, Inversion. Wikipedia, q-Pochhammer symbol. - Paul Muljadi, Jan 18 2011 Index entries for "core" sequences. FORMULA Largest coefficient of (1)(x+1)(x^2+x+1)...(x^(n-1) + ... + x + 1). - David W. Wilson The number of terms is given in A000124. a(n+1)/a(n) = n - 1/2 + O(1/n^{1-epsilon}) as n --> infinity (compare with A008302, A181609, A001147). - Mikhail Gaichenkov, Apr 11 2014 Asymptotics (Mikhail Gaichenkov, 2010): a(n) ~ 6 * n^(n-1) / exp(n). - Vaclav Kotesovec, May 17 2015 EXAMPLE From Joerg Arndt, Jan 16 2011: (Start) a(4) = 6 because the among the permutations of 4 elements those with 3 inversions are the most frequent and appear 6 times: [inv. table] [permutation] number of inversions 0: [ 0 0 0 ] [ 0 1 2 3 ] 0 1: [ 1 0 0 ] [ 1 0 2 3 ] 1 2: [ 0 1 0 ] [ 0 2 1 3 ] 1 3: [ 1 1 0 ] [ 2 0 1 3 ] 2 4: [ 0 2 0 ] [ 1 2 0 3 ] 2 5: [ 1 2 0 ] [ 2 1 0 3 ] 3 (*) 6: [ 0 0 1 ] [ 0 1 3 2 ] 1 7: [ 1 0 1 ] [ 1 0 3 2 ] 2 8: [ 0 1 1 ] [ 0 3 1 2 ] 2 9: [ 1 1 1 ] [ 3 0 1 2 ] 3 (*) 10: [ 0 2 1 ] [ 1 3 0 2 ] 3 (*) 11: [ 1 2 1 ] [ 3 1 0 2 ] 4 12: [ 0 0 2 ] [ 0 2 3 1 ] 2 13: [ 1 0 2 ] [ 2 0 3 1 ] 3 (*) 14: [ 0 1 2 ] [ 0 3 2 1 ] 3 (*) 15: [ 1 1 2 ] [ 3 0 2 1 ] 4 16: [ 0 2 2 ] [ 2 3 0 1 ] 4 17: [ 1 2 2 ] [ 3 2 0 1 ] 5 18: [ 0 0 3 ] [ 1 2 3 0 ] 3 (*) 19: [ 1 0 3 ] [ 2 1 3 0 ] 4 20: [ 0 1 3 ] [ 1 3 2 0 ] 4 21: [ 1 1 3 ] [ 3 1 2 0 ] 5 22: [ 0 2 3 ] [ 2 3 1 0 ] 5 23: [ 1 2 3 ] [ 3 2 1 0 ] 6 The statistics are reflected by the coefficients of the polynomial (1+x)*(1+x+x^2)*(1+x+x^2+x^3) == x^6 + 3*x^5 + 5*x^4 + 6*x^3 + 5*x^2 + 3*x^1 + 1*x^0 There is 1 permutation (the identity) with 0 inversions, 3 permutations with 1 inversion, 5 with 2 inversions, 6 with 3 inversions (the most frequent, marked with (*) ), 5 with 4 inversions, 3 with 5 inversions, and one with 6 inversions. (End) G.f. = x + x^2 + 2*x^3 + 6*x^4 + 22*x^5 + 101*x^6 + 573*x^7 + 3836*x^8 + ... MAPLE f := 1: for n from 0 to 40 do f := f*add(x^i, i=0..n): s := series(f, x, n*(n+1)/2+1): m := max(coeff(s, x, j) \$ j=0..n*(n+1)/2): printf(`%d, `, m) od: # James A. Sellers, Dec 07 2000 [offset is off by 1 - N. J. A. Sloane, May 23 2006] P:= : a:= 1: for n from 2 to 100 do P:= expand(P * add(x^j, j=0..n-1)); a[n]:= max(eval(convert(P, list), x=1)); od: seq(a[i], i=1..100); # Robert Israel, Dec 14 2014 MATHEMATICA f[n_] := Max@ CoefficientList[ Expand@ Product[ Sum[x^i, {i, 0, j}], {j, n-1}], x]; Array[f, 20] Flatten[{1, 1, Table[Coefficient[Expand[Product[Sum[x^k, {k, 0, m-1}], {m, 1, n}]], x^Floor[n*(n-1)/4]], {n, 3, 20}]}] (* Vaclav Kotesovec, May 13 2016 *) Table[SeriesCoefficient[QPochhammer[x, x, n]/(1-x)^n, {x, 0, Floor[n*(n-1)/4]}], {n, 1, 20}] (* Vaclav Kotesovec, May 13 2016 *) PROG (PARI) a(n)= /* return largest coefficient in product (1)(x+1)(x^2+x+1)...(x^(n-1)+...+x+1) */ { local(p, v); p=prod(k=1, n-1, sum(j=0, k, x^j)); /* polynomial */ v=Vec(p); /* vector of coefficients */ v=vecsort(v); /* sort so largest is last element */ return(v[#v]); /* return last == largest */ } vector(22, n, a(n)) /* Joerg Arndt, Jan 16 2011 */ (Magma) /* based on David W. Wilson's formula */ PS:=PowerSeriesRing(Integers()); [ Max(Coefficients(&*[&+[ x^i: i in [0..j] ]: j in [0..n-1] ])): n in [1..21] ]; // Klaus Brockhaus, Jan 18 2011 (PARI) {a(n) = if( n<0, 0, vecmax( Vec( prod(k=1, n, 1 - x^k) / (1 - x)^n)))}; /* Michael Somos, Apr 21 2014 */ (Python) from math import prod from sympy import Poly from sympy.abc import x def A000140(n): return 1 if n == 1 else max(Poly(prod(sum(x**i for i in range(j+1)) for j in range(n))).all_coeffs()) # Chai Wah Wu, Feb 02 2022 CROSSREFS Row maxima of A008302. Odd terms are A186888. Cf. A000124, A001147, A128566, A181609. Sequence in context: A012268 A009655 A002772 * A079263 A129815 A345194 Adjacent sequences: A000137 A000138 A000139 * A000141 A000142 A000143 KEYWORD nonn,easy,core,nice AUTHOR N. J. A. Sloane EXTENSIONS Edited by N. J. A. Sloane, Mar 05 2011 STATUS approved

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Last modified November 30 18:14 EST 2023. Contains 367461 sequences. (Running on oeis4.)