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 A189073 Triangle read by rows: T(n,k) is the number of inversions in k-compositions of n for n >= 3, 2 <= k <= n-1. 2
 1, 1, 3, 2, 6, 6, 2, 12, 18, 10, 3, 18, 42, 40, 15, 3, 27, 78, 110, 75, 21, 4, 36, 132, 240, 240, 126, 28, 4, 48, 204, 460, 600, 462, 196, 36, 5, 60, 300, 800, 1290, 1302, 812, 288, 45, 5, 75, 420, 1300, 2490, 3108, 2548, 1332, 405, 55, 6, 90, 570, 2000, 4440, 6594, 6692, 4608, 2070, 550, 66 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 3,3 COMMENTS The Heibach et al. reference has a table for n <= 14. LINKS Alois P. Heinz, Rows n = 3..143, flattened S. Heubach, A. Knopfmacher, M. E. Mays and A. Munagi, Inversions in Compositions of Integers, to appear in Quaestiones Mathematicae. FORMULA G.f.: (1-x)*x^3/((1+x)*(1-x-y*x)^3). - Geoffrey Critzer, Mar 19 2014 EXAMPLE Triangle begins: 1; 1, 3; 2, 6, 6; 2, 12, 18, 10; 3, 18, 42, 40, 15; 3, 27, 78, 110, 75, 21; 4, 36, 132, 240, 240, 126, 28; 4, 48, 204, 460, 600, 462, 196, 36; 5, 60, 300, 800, 1290, 1302, 812, 288, 45; 5, 75, 420, 1300, 2490, 3108, 2548, 1332, 405, 55; 6, 90, 570, 2000, 4440, 6594, 6692, 4608, 2070, 550, 66; ... T(5,3) = 6 because we have: 3+1+1, 1+3+1, 1+1+3, 2+2+1, 2+1+2, 1+2+2 having 2,1,0,2,1,0 inversions respectively. - Geoffrey Critzer, Mar 19 2014 MAPLE T:= proc(n, k) option remember; if k=2 then floor((n-1)/2) elif k>=n then 0 else T(n-1, k) +k/(k-2) *T(n-1, k-1) fi end: seq(seq(T(n, k), k=2..n-1), n=3..13); # Alois P. Heinz, Apr 17 2011 MATHEMATICA T[n_, k_] := T[n, k] = Which[k == 2, Floor[(n-1)/2], k >= n, 0, True, T[n-1, k] + k/(k-2)*T[n-1, k-1]]; Table[Table[T[n, k], {k, 2, n-1}], {n, 3, 13}] // Flatten (* Jean-François Alcover, Jan 14 2014, after Alois P. Heinz *) CROSSREFS Row sums are A189052. The first column is A004526(n-1). Diagonal is A000217(n-2). Lower diagonal is A002411(n-3). 2nd lower diagonal is A001621(n-4). Sequence in context: A293549 A306443 A336518 * A107271 A196565 A104633 Adjacent sequences: A189070 A189071 A189072 * A189074 A189075 A189076 KEYWORD nonn,tabl AUTHOR N. J. A. Sloane, Apr 16 2011 STATUS approved

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Last modified November 28 09:27 EST 2023. Contains 367412 sequences. (Running on oeis4.)