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 A064315 Triangle of number of permutations by length of shortest ascending run. 14
 1, 1, 1, 5, 0, 1, 18, 5, 0, 1, 101, 18, 0, 0, 1, 611, 89, 19, 0, 0, 1, 4452, 519, 68, 0, 0, 0, 1, 36287, 3853, 110, 69, 0, 0, 0, 1, 333395, 27555, 1679, 250, 0, 0, 0, 0, 1, 3382758, 233431, 11941, 418, 251, 0, 0, 0, 0, 1, 37688597, 2167152, 59470, 658, 922, 0, 0, 0, 0, 0, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS T(2*n,n) = binomial(2*n,n)-1 = A030662(n). Sum_{k=1..n} k * T(n,k) = A064316(n). LINKS Alois P. Heinz, Rows n = 1..100, flattened D. W. Wilson, Extended tables for A008304 and A064315 FORMULA Sequence (1, 3, 2, 5, 4) has ascending runs (1, 3), (2, 5), (4), the shortest is length 1. Of all permutations of (1, 2, 3, 4, 5), T(5,1) = 101 have shortest ascending run of length 1. EXAMPLE Triangle begins:       1;       1,    1;       5,    0,   1;      18,    5,   0,  1;     101,   18,   0,  0,  1;     611,   89,  19,  0,  0, 1;    4452,  519,  68,  0,  0, 0, 1,   36287, 3853, 110, 69,  0, 0, 0, 1; MAPLE A:= proc(n, k) option remember; local b; b:=       proc(u, o, t) option remember; `if`(t+o<=k, (u+o)!,         add(b(u+i-1, o-i, min(k, t)+1), i=1..o)+         `if`(t<=k, u*(u+o-1)!, add(b(u-i, o+i-1, 1), i=1..u)))       end: forget(b):       add(b(j-1, n-j, 1), j=1..n)     end: T:= (n, k)-> A(n, k) -A(n, k-1): seq(seq(T(n, k), k=1..n), n=1..12); # Alois P. Heinz, Aug 29 2013 MATHEMATICA A[n_, k_] := A[n, k] = Module[{b}, b[u_, o_, t_] := b[u, o, t] = If[t+o <= k, (u+o)!, Sum[b[u+i-1, o-i, Min[k, t]+1], {i, 1, o}] + If[t <= k, u*(u+o-1)!, Sum[ b[u-i, o+i-1, 1], {i, 1, u}]]]; Sum[b[j-1, n-j, 1], {j, 1, n}]]; T[n_, k_] := A[n, k] - A[n, k-1]; Table[Table[T[n, k], {k, 1, n}], {n, 1, 12}] // Flatten (* Jean-François Alcover, Jan 28 2015, after Alois P. Heinz *) CROSSREFS Row sums give: A000142. Columns k=1-10 give: A228614, A185652, A228670, A228671, A228672, A228673, A228674, A228675, A228676, A228677. Cf. A030662. Sequence in context: A291774 A222061 A345453 * A227322 A216718 A184180 Adjacent sequences:  A064312 A064313 A064314 * A064316 A064317 A064318 KEYWORD nonn,tabl AUTHOR David W. Wilson, Sep 07 2001 STATUS approved

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Last modified July 27 15:49 EDT 2021. Contains 346308 sequences. (Running on oeis4.)