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 A008304 Triangle read by rows: T(n,k) (n>=1; 1<=k<=n) is the number of permutations of [n] in which the longest increasing run has length k. 27
 1, 1, 1, 1, 4, 1, 1, 16, 6, 1, 1, 69, 41, 8, 1, 1, 348, 293, 67, 10, 1, 1, 2016, 2309, 602, 99, 12, 1, 1, 13357, 19975, 5811, 1024, 137, 14, 1, 1, 99376, 189524, 60875, 11304, 1602, 181, 16, 1, 1, 822040, 1960041, 690729, 133669, 19710, 2360, 231, 18, 1, 1, 7477161 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS Row n has n terms. REFERENCES F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 261, Table 7.4.1. LINKS Alois P. Heinz, Rows n = 1..141, flattened Max A. Alekseyev, On the number of permutations with bounded run lengths, arXiv preprint arXiv:1205.4581, 2012. - From N. J. A. Sloane, Oct 23 2012 D. W. Wilson, Extended tables for A008304 and A064315 FORMULA E.g.f. of column k: 1/Sum_{n>=0} ((k+1)*n+1-x)*x^((k+1)*n)/((k+1)*n+1)! - 1/Sum_{n>=0} (k*n+1-x)*x^(k*n)/(k*n+1)!. - Alois P. Heinz, Oct 13 2013 T(n,k) = A122843(n,k) for k > n/2. - Alois P. Heinz, Oct 17 2013 EXAMPLE 1; 1,   1; 1,   4,   1; 1,  16,   6,  1; 1,  69,  41,  8,  1; 1, 348, 293, 67, 10,  1; ... T(3,2) = 4 because we have (13)2, 2(13), (23)1, 3(12), where the parentheses surround runs of length 2. MAPLE b:= proc(u, o, t, k) option remember; `if`(t=k, (u+o)!,       `if`(max(t, u)+o b(0, n, 0, k) -b(0, n, 0, k+1): seq(seq(T(n, k), k=1..n), n=1..15);  # Alois P. Heinz, Oct 16 2013 MATHEMATICA b[u_, o_, t_, k_] := b[u, o, t, k] = If[t == k, (u + o)!, If[Max[t, u]+o < k, 0, Sum[b[u+j-1, o-j, t+1, k], {j, 1, o}] + Sum[b[u-j, o+j-1, 1, k], {j, 1, u}]]]; T[n_, k_] := b[0, n, 0, k] - b[0, n, 0, k+1]; Table[Table[T[n, k], {k, 1, n}], {n, 1, 15}] // Flatten (* Jean-François Alcover, Jan 10 2014, translated from Alois P. Heinz's Maple code *) ' (*additional code*) nn=12; a[r_]:=Apply[Plus, Table[Normal[Series[y x^(r+1)/(1-Sum[y x^i, {i, 1, r}]), {x, 0, nn}]][[n]]/(n+r)!, {n, 1, nn-r}]]/.y->-1; Map[Select[#, #>0&]&, Transpose[Prepend[Table[Drop[Range[0, nn]! CoefficientList[Series[1/(1-x-a[n+1])-1/(1-x-a[n]), {x, 0, nn}], x], 1], {n, 1, 8}], Table[1, {nn}]]]]//Grid (* Geoffrey Critzer, Feb 25 2014 *) CROSSREFS Row sums give A000142. Sum_{k=1..n} k*T(n,k) = A064314(n). Cf. A064315. Columns k=1-10 give: A000012, A000303, A000402, A000434, A000456, A000467, A230055, A230234, A230235, A230236. T(2n+j,n+j) for j=0-10 gives: A230341, A230251, A230342, A230343, A230344, A230345, A230346, A230347, A230348, A230349, A230350. Sequence in context: A155826 A010320 A152571 * A203846 A118185 A176483 Adjacent sequences:  A008301 A008302 A008303 * A008305 A008306 A008307 KEYWORD nonn,tabl AUTHOR EXTENSIONS More terms from David W. Wilson, Sep 07 2001 Better description from Emeric Deutsch, May 08 2004 STATUS approved

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