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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-Fran├žois 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: A221800 A291774 A222061 * 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 April 10 02:39 EDT 2020. Contains 333392 sequences. (Running on oeis4.)