

A008302


Triangle of Mahonian numbers T(n,k): coefficients in expansion of Product_{i=0..n1} (1 + x + ... + x^i), where k ranges from 0 to A000217(n1).


88



1, 1, 1, 1, 2, 2, 1, 1, 3, 5, 6, 5, 3, 1, 1, 4, 9, 15, 20, 22, 20, 15, 9, 4, 1, 1, 5, 14, 29, 49, 71, 90, 101, 101, 90, 71, 49, 29, 14, 5, 1, 1, 6, 20, 49, 98, 169, 259, 359, 455, 531, 573, 573, 531, 455, 359, 259, 169, 98, 49, 20, 6, 1
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OFFSET

1,5


COMMENTS

T(n,k) = number of permutations of {1..n} with k inversions.
nth row gives growth series for symmetric group S_n with respect to transpositions (1,2), (2,3), ..., (n1,n).
T(n,k) = number of permutations of (1,2,...,n) having disorder equal to k. The disorder of a permutation p of (1,2,...,n) is defined in the following manner. We scan p from left to right as often as necessary until all its elements are removed in increasing order, scoring one point for each occasion on which an element is passed over and not removed. The disorder of p is the number of points scored by the end of the scanning and removal process. For example, the disorder of (3,5,2,1,4) is 8, since on the first scan, 3,5,2 and 4 are passed over, on the second, 3,5 and 4 and on the third scan, 5 is once again not removed.  Emeric Deutsch, Jun 09 2004
T(n,k)=number of permutations p=(p(1),...p(n)) of {1..n} such that Sum(i: p(i)>p(i+1))=k (k is called the Major index of p). Example: T(3,0)=1, T(3,1)=2,T(3,2)=2,T(3,3)=1 because the Major indices of the permutations (1,2,3), (2,1,3),(3,1,2),(1,3,2),(2,3,1) and (3,2,1) are 0,1,1,2,2 and 3, respectively.  Emeric Deutsch, Aug 17 2004
T(n,k) = number of 2 X c matrices with column totals 1,2,3,...,n and row totals k and binomial(n+1,2)  k.  Mitch Harris, Jan 13 2006
T(n,k) is the number of permutations p of {1,2,...,n} for which den(p)=k. Here den is the Denert statistic, defined in the following way: let p=p(1)p(2)...p(n) be a permutation of {1,2,...,n}; if p(i)>i, then we say that i is an excedance of p; let i_1 < i_2 < ... < i_k be the excedances of p and let j_1 < j_2 < ... < j_{nk} be the nonexcedances of p; let Exc(p) = p(i_1)p(i_2)...p(i_k), Nexc(p)=p(j_1)p(j_2)...p(j_{nk}); then, by definition den(p)=i_1 + i_2 + ... + i_k + inv(Exc(p)) + inv(Nexc(p)), where inv denotes "number of inversions". Example: T(4,5)=3 because we have 1342, 3241 and 4321. We show that den(4321)=5: the excedances are 1 and 2; Exc(4321)=43, Nexc(4321)=21; now den(4321)=1+2+inv(43)+inv(21)=3+1+1=5.  Emeric Deutsch, Oct 29 2008
T(n,k) is the number of size k submultisets of the multiset {1,2,2,3,3,3,...,n1} (which contains i copies of i for 0 < i < n).
The limit of products of the numbers of fixed necklaces of length n composed of beads of types N(n,a), n> infinity, is the generating function for inversions (we must exclude one unimportant factor a^n/n!). The error is < a^n/n!*O(1/n^(1/2epsilon)). See Gaichenkov link.  Mikhail Gaichenkov, Aug 27 2012
The number of ways to distribute k1 indistinguishable balls into n1 boxes of capacity 1,2,3,...n1.  Andrew Woods, Sep 26 2012
Partial sums of rows give triangle A161169.  András Salamon, Feb 16 2013
The number of permutations of n that require k pair swaps in the bubble sort to sort them into the natural 1,2,...,n order.  R. J. Mathar, May 04 2013
Also series coefficients of qfactorial [n]_q !  see Mathematica line.  Wouter Meeussen, Jul 12 2014
Comments from Mikhail Gaichenkov, Aug 16 2016 (Start)
Following asymptotic expansions in the Central Limit Theorem developed by Valentin V. Petrov, the cumulative distribution function of these numbers, CDF_N(x), is equal to the CDF of the normal distribution  (0.06/sqrt(2PI))*exp(x^2/2)(x^33x)*(6N^3+21N^2+31N+31)/(N(2N+5)^2(N1)+O(1/N^2).
This can be written as: CDF of the normal distribution (0.09/(N*sqrt(2PI)))*exp(x^2/2)He_3(x)+O(1/N^2), N>1, natural numbers (Gaichenkov, private research).
According to B. H. Margolius, Permutations with inversions, J. Integ. Seqs. Vol. 4 (2001), #01.2.4 ‘the unimodal behavior of the inversion numbers suggests that the number of inversions in a random permutation may be asymptotically normal’ https://cs.uwaterloo.ca/journals/JIS/VOL4/MARGOLIUS/inversions.pdf.
Moreover, E. BenNaim (Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory), ‘On the Mixing of Diffusing Particles’ ( 13 Oct 2010), states that the MAHONIAN DISTRIBUTION becomes a function of a single variable for large numbers of element, i.e. the probability distribution function is normal. E. BenNaim gives some examples here: https://arxiv.org/pdf/1010.2563v1.pdf.
To be more precise the expansion of the distribution is presented for a finite number of elements (or particles in terms of E. BenNaim’s article). The distribution tends to the normal distribution for an infinite numbers of elements.
(End)


REFERENCES

M. Bona, Combinatorics of permutations, Chapman & Hall/CRC, Boca Raton, Florida, 2004 (p. 52).
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 240.
F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 241.
P. de la Harpe, Topics in Geometric Group Theory, Univ. Chicago Press, 2000, p. 163, top display.
E. Netto, Lehrbuch der Combinatorik. 2nd ed., Teubner, Leipzig, 1927, p. 96.
Valentin V. Petrov, Sums of Independent Random Variables, Springer Berlin Heidelberg, 1975, p. 134.
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 1, 1999; see Corollary 1.3.10, p. 21.


LINKS

T. D. Noe, Rows n=0..30 of triangle, flattened
F. Brglez, Of ndimensional Dice, Combinatorial Optimization, and Reproducible Research: An Introduction, Elektrotehniski Vestnik, 78(4): 181192, 2011.
L. Carlitz, qBernoulli numbers and polynomials, Duke Math. J. Volume 15, Number 4 (1948), 9871000.
E. Deutsch, Problem 10975: Enumeration of Permutations by Disorder, Amer. Math. Monthly, 111 (2004), 541.
FindStat  Combinatorial Statistic Finder, The Denert index of a permutation, The major index of a permutation, The number of inversions of a permutation, The sorting index of a permutation
D. Foata, Distributions eulériennes et mahoniennes sur le groupe des permutations, pp. 2749 of M. Aigner, editor, Higher Combinatorics, Reidel, Dordrecht, Holland, 1977.
D. Foata and D. Zeilberger, Denert's permutation statistic is indeed EulerMahonian, Studies in Appl. Math., 83(1990),3159. [From Emeric Deutsch, Oct 29 2008]
Mikhail Gaichenkov, Necklaces and the generating function for inversions
Mikhail Gaichenkov, Pros and cons of probability model for permutations
Amy Grady, Sorting index and MahonianStirling pairs for labeled forests, Clemson University, July 10, 2014.
GuoNiu Han, Une nouvelle bijection pour la statistique de Denert, C. R. Acad. Sci. Paris, Ser. I, 310(1990),493496.
Stuart A. Hannah, Sieved Enumeration of Interval Orders and Other Fishburn Structures, arXiv:1502.05340 [math.CO], (18February2015).
Y.H. He, C. Matti, C. Sun, The Scattering Variety, arXiv preprint arXiv:1403.6833 [hepth], 2014.
E. Irurozki, Sampling and learning distancebased probability models for permutation spaces, PhD Dissertation, Department of Computer Science and Artificial Intelligence of the University of the Basque Country, 2015.
E. Irurozki, B. Calvo, J. A. Lozano, An R package for permutations, Mallows and Generalized Mallows models, 2014.
M. Janjic, A Generating Function for Numbers of Insets, Journal of Integer Sequences, 17, 2014, #14.9.7.
J. A. Koziol, Sums of ranking differences and inversion numbers for method discrimination, Journal of Chemometrics, 27 (2013): 165169. doi: 10.1002/cem.2504
B. H. Margolius, Permutations with inversions, J. Integ. Seqs. Vol. 4 (2001), #01.2.4.
J. L. Martin and J. D. Wagner, On the Spectra of Simplicial Rook Graphs, arXiv preprint arXiv:1209.3493 [math.CO], 2012.  From N. J. A. Sloane, Dec 27 2012
A. Mendes, A note on alternating permutations, Amer. Math. Monthly, 114 (2007), 437440.
R. H. Moritz and R. C. Williams, A cointossing problem and some related combinatorics, Math. Mag., 61 (1988), 2429.
E. Netto, Lehrbuch der Combinatorik, Chapter 4, annotated scanned copy of pages 9299 only.
E. Netto, Lehrbuch der Combinatorik, Chapter 4, annotated scanned copy of pages 9299 only.
Michal Opler, Major index distribution over permutation classes, arXiv:1505.07135 [math.CO], 2015.
Svetlana Poznanovic, The sorting index and equidistribution of setvalued statistics over restricted permutations, Journal of Combinatorial Theory, Series A, 125 (2014), 254272.
R. P. Stanley, Binomial posets, Moebius inversion and permutation enumeration, J. Combinat. Theory, A 20 (1976), 336356.
A. Waksman, On the complexity of inversions, IEEE Trans. Computers, 19 (1970), 12251226. See Table II.
Eric W. Weisstein, Mathworld: Necklace
Eric W. Weisstein, Mathworld: Irreducible Polynomial
Thomas Wieder, Comments on A008302
Wikipedia, Major index


FORMULA

Comtet and MoritzWilliams give recurrences.
Mendes and Stanley give g.f.'s.
G.f.: Product_{j=1..n} (1x^j)/(1x) = Sum_{k=0..M} T{n, k} x^k, where M = n*(n1)/2.
From Andrew Woods, Sep 26 2012: (Start)
T(1, 1) = 1, T(1, k != 1) = 0,
T(n, k) = Sum_{j=0..n1} T(n1, kj),
T(n, k) = T(n, k1) + T(n1, k)  T(n1, kn). (End)


EXAMPLE

1; 1+x; (1+x)*(1+x+x^2) = 1+2*x+2*x^2+x^3; etc.
Triangle begins:
1;
1, 1;
1, 2, 2, 1;
1, 3, 5, 6, 5, 3, 1;
1, 4, 9, 15, 20, 22, 20, 15, 9, 4, 1;
1, 5, 14, 29, 49, 71, 90, 101, 101, 90, 71, 49, 29, 14, 5, 1;
1, 6, 20, 49, 98, 169, 259, 359, 455, 531, 573, 573, 531, 455, 359, 259, 169, 98, 49, 20, 6, 1
...


MAPLE

g := proc(n, k) option remember; if k=0 then return(1) else if (n=1 and k=1) then return(0) else if (k<0 or k>binomial(n, 2)) then return(0) else g(n1, k)+g(n, k1)g(n1, kn) end if end if end if end proc; # Barbara Haas Margolius (margolius(AT)math.csuohio.edu), May 31 2001
BB:=j>1+sum(t^i, i=1..j): for n from 1 to 8 do Z[n]:=sort(expand(simplify(product(BB(j), j=0..n2)))) od: for n from 1 to 8 do seq(coeff(Z[n], t, j), j=0..(n1)*(n2)/2) od; # Zerinvary Lajos, Apr 13 2007


MATHEMATICA

f[n_] := CoefficientList[ Expand@ Product[ Sum[x^i, {i, 0, j}], {j, n}], x]; Flatten[Array[f, 8, 0]]
nn = 25; T[1, 1] = 1; T[n_, 1] = 0; T[n_, k_] := T[n, k] = Sum[T[n  i, k  1], {i, 1, k  1}]; MatrixForm[Table[T[n, k], {n, nn}, {k, nn}]] (* Mats Granvik and Roger L. Bagula, Jun 19 2011 *)
alternatively (versions 7 and up):
Table[CoefficientList[Series[QFactorial[n, q], {q, 0, n(n1)/2}], q], {n, 9}] (* Wouter Meeussen, Jul 12 2014 *)


PROG

(Sage)
from sage.combinat.q_analogues import q_factorial
for n in (1..6): print q_factorial(n).list() # Peter Luschny, Jul 18 2016


CROSSREFS

Diagonals give A000707, A001892, A001893, A001894, A005283, A005284, A005285, A005286, A005287, A005288, A242656, A242657.
Rowmaxima: A000140, truncated table: A060701, row sums: A000142.
A161436 is one of the rows.
A001809 gives total Denert index of all permutations.
Cf. A139365.
Sequence in context: A076263 A272689 * A274887 A131791 A010358 A155865
Adjacent sequences: A008299 A008300 A008301 * A008303 A008304 A008305


KEYWORD

easy,tabf,nonn,nice,look


AUTHOR

N. J. A. Sloane


EXTENSIONS

There were some mistaken edits to this entry (inclusion of an initial 1, etc.) which I undid.  N. J. A. Sloane, Nov 30 2009


STATUS

approved



