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 A129178 Triangle read by rows: T(n,k) is the number of permutations p of {1,2,...,n} such that invc(p)=k (n >= 0; 0 <= k <= (n-1)(n-2)/2) (see comment for invc definition). 5
 1, 1, 2, 4, 2, 8, 8, 6, 2, 16, 24, 28, 26, 16, 8, 2, 32, 64, 96, 120, 126, 110, 82, 52, 26, 10, 2, 64, 160, 288, 432, 564, 658, 680, 638, 542, 416, 284, 172, 90, 38, 12, 2, 128, 384, 800, 1376, 2072, 2824, 3526, 4058, 4344, 4346, 4066, 3562, 2912, 2218, 1566, 1016, 598 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS invc(p) is defined (by Carlitz) in the following way: express p in standard cycle form (i.e., cycles ordered by increasing smallest elements with each cycle written with its smallest element in the first position), then remove the parentheses and count the inversions in the obtained word. Row n has 1+(n-1)*(n-2)/2 - delta_{0,n} terms. Row sums are the factorials (A000142). T(n,0) = 2^(n-1) = A011782(n) = A000079(n-1). T(n,1) = (n-2)*2^(n-2) = A036289(n-2) for n>=2. T(n,k) = A121552(n,n+k). It appears that Sum_{k>=0} k*T(n,k) = A126673(n). REFERENCES L. Carlitz, Generalized Stirling numbers, Combinatorial Analysis Notes, Duke University, 1968, 1-7. LINKS Alois P. Heinz, Rows n = 0..50, flattened Toufik Mansour, Mark Shattuck, A q-analog of the hyperharmonic numbers, Afrika Matematika 25.1 (2014): 147-160. M. Shattuck, Parity theorems for statistics on permutations and Catalan words, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 5, Paper A07, 2005. FORMULA Generating polynomial of row n is P[n](t) = 2*(2+t)*(2+t+t^2)*...*(2 + t + t^2 + ... + t^(n-2)) for n >= 3, P[1](t)=1, P[2](t)=2. EXAMPLE T(3,0)=4, T(3,1)=2 because we have 123=(1)(2)(3), 132=(1)(23), 213=(12)(3), 231=(123) with the resulting word (namely 123) having 0 inversions and 312=(132) and (321)=(13)(2) with the resulting word (namely 132) having 1 inversion. Triangle starts:    1;    1;    2;    4,   2;    8,   8,   6,   2;   16,  24,  28,  26,  16,   8,   2;   32,  64,  96, 120, 126, 110,  82,  52,  26,  10,  2; MAPLE s:=j->2+sum(t^i, i=1..j): for n from 0 to 9 do P[n]:=sort(expand(simplify(product(s(j), j=0..n-2)))) od: for n from 0 to 9 do seq(coeff(P[n], t, j), j=0..degree(P[n])) od;  # yields sequence in triangular form MATHEMATICA nMax = 9; s[j_] := 2 + Sum[t^i, {i, 1, j}]; P[0] = P[1] = 1; P[2] = 2; For[ n = 3, n <= nMax, n++, P[n] = Sort[Expand[Simplify[Product[s[j], {j, 0, n-2}]]]]]; Table[Coefficient[P[n], t, j], {n, 0, nMax}, {j, 0, Exponent[ P[n], t]}] // Flatten (* Jean-François Alcover, Jan 24 2017, adapted from Maple *) CROSSREFS Cf. A000142, A011782, A000079, A036289, A121552, A126673. Sequence in context: A287879 A336093 A304213 * A152874 A324716 A328378 Adjacent sequences:  A129175 A129176 A129177 * A129179 A129180 A129181 KEYWORD nonn,tabf AUTHOR Emeric Deutsch, Apr 11 2007 EXTENSIONS One term for row n=0 prepended by Alois P. Heinz, Dec 16 2016 STATUS approved

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Last modified April 11 03:58 EDT 2021. Contains 342886 sequences. (Running on oeis4.)