0,6

T(n,k) = number of permutations of {1..n} with k inversions.

n-th row gives growth series for symmetric group S_n with respect to transpositions (1,2), (2,3), ..., (n-1,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 (deutsch(AT)duke.poly.edu), 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 (deutsch(AT)duke.poly.edu), Aug 17 2004

T(n,k) = number of 2 x c matrices with column totals 1,2,3,...,n and row totals k and (n+1 choose 2) - k. - Mitch Harris (Harris.Mitchell(AT)mgh.harvard.edu), 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_{n-k} be the non-excedances of p; let Exc(p) = p(i_1)p(i_2)...p(i_k), Nexc(p)=p(j_1)p(j_2)...p(j_{n-k}); 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. [From Emeric Deutsch (deutsch(AT)duke.poly.edu), Oct 29 2008]

M. Bona, Combinatorics of permutations, Chapman & Hall/CRC, Boca Raton, Florida, 2004 (p. 52).

L. Carlitz, q-Bernoulli numbers and polynomials, Duke Math. J. Volume 15, Number 4 (1948), 987-1000. See http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.dmj/1077475200 [from Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 06 2009]

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.

E. Deutsch, Problem 10975, Amer. Math. Monthly, 111 (2004), 541.

D. Foata, Distributions eule'riennes et mahoniennes sur le group des permutations, pp. 27-49 of M. Aigner, editor, Higher Combinatorics, Reidel, Dordrecht, Holland, 1977.

D. Foata and D. Zeilberger, Denert's permutation statistic is indeed Euler-Mahonian, Studies in Appl. Math., 83(1990),31-59. [From Emeric Deutsch (deutsch(AT)duke.poly.edu), Oct 29 2008]

Guo-Niu Han, Une nouvelle bijection pour la statistique de Denert, C. R. Acad. Sci. Paris, Ser. I, 310(1990),493-496. [From Emeric Deutsch (deutsch(AT)duke.poly.edu), Oct 29 2008]

P. de la Harpe, Topics in Geometric Group Theory, Univ. Chicago Press, 2000, p. 163, top display.

A. Mendes, A note on alternating permutations, Amer. Math. Monthly, 114 (2007), 437-440.

R. H. Moritz and R. C. Williams, A coin-tossing problem and some related combinatorics, Math. Mag., 61 (1988), 24-29.

E. Netto, Lehrbuch der Combinatorik. 2nd ed., Teubner, Leipzig, 1927, p. 96.

R. P. Stanley, Binomial posets, Moebius inversion and permutation enumeration, J. Combinat. Theory, A 20 (1976), 336-356.

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 1, 1999; see Corollary 1.3.10, p. 21.

T. D. Noe, Rows n=0..30 of triangle, flattened

B. H. Margolius, Permutations with inversions, J. Integ. Seqs. Vol. 4 (2001), #01.2.4.

Thomas Wieder, Comments on A008302

Comtet and Moritz-Williams give recurrences.

Mendes and Stanley give g.f.'s.

G.f.: Product_{j=1..n} (1-x^j)/(1-x) = Sum_{k=1..M} T{n, k} x^k, where M = n(n-1)/2.

1; 1+x; (1+x)(1+x+x^2) = 1+2x+2x^2+x^3; etc.

{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},

{1, 7, 27, 76, 174, 343, 602, 961, 1415, 1940, 2493, 3017, 3450, 3736, 3836, 3736, 3450, 3017, 2493, 1940, 1415, 961, 602, 343, 174, 76, 27, 7, 1},

{1, 8, 35, 111, 285, 628, 1230, 2191, 3606, 5545, 8031, 11021, 14395, 17957, 21450, 24584, 27073, 28675, 29228, 28675, 27073, 24584, 21450, 17957, 14395, 11021, 8031, 5545, 3606, 2191, 1230, 628, 285, 111, 35, 8, 1},

{1, 9, 44, 155, 440, 1068, 2298, 4489, 8095, 13640, 21670, 32683, 47043, 64889, 86054, 110010, 135853, 162337, 187959, 211089, 230131, 243694, 250749, 250749, 243694, 230131, 211089, 187959, 162337, 135853, 110010, 86054, 64889, 47043, 32683, 21670, 13640, 8095, 4489, 2298, 1068, 440, 155, 44, 9, 1}

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(n-1, k)+g(n, k-1)-g(n-1, k-n) end if end if end if end proc;

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..n-2)))) od: for n from 1 to 8 do seq(coeff(Z[n], t, j), j=0..(n-1)*(n-2)/2) od; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 13 2007

p[x_, n_] = Product[(1 - x^k)/(1 - x), {k, 1, n}];

Table[FullSimplify[ExpandAll[p[x, n]]], {n, 0, 10}];

Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 0, 10}];

Flatten[%] [From Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 06 2009]

Diagonals give A000707, A001892, A001893, A001894, A005283, A005284, A005285, A005286, A005287, A005288.

Row-maxima: A000140, Truncated table: A060701

Sequence in context: A060351 A076037 A076263 this_sequence A131791 A010358 A155865

Adjacent sequences: A008299 A008300 A008301 this_sequence A008303 A008304 A008305

easy,tabf,nonn,nice,new

N. J. A. Sloane (njas(AT)research.att.com).

Maple code from Barbara Haas Margolius (margolius(AT)math.csuohio.edu) 5/31/01

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