A225960 -id:A225960 - cp's OEIS frontend

A260957 Coefficients in asymptotic expansion of sequence A225960.

1, -1, 0, 2, 11, 97, 1105, 13905, 189633, 2803873, 44875599, 774931369, 14385283720, 286010393240, 6069158797378, 137001560841500, 3279733485742038, 83029219746406610, 2216965307395350215, 62282348739259200015, 1836840301301564071196, 56750954571528670989664

Offset: 0

Vaclav Kotesovec, Aug 05 2015

sign

			A225960(n)/(2*(n-1)!) ~ 1 - 1/n + 2/n^3 + 11/n^4 + 97/n^5 + 1105/n^6 + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..132

Cf. A225960, A256168.

a(k) ~ k! / (2 * (log(2))^(k+1)). - Vaclav Kotesovec, Aug 05 2015

A186373 Triangle read by rows: T(n,k) is the number of permutations of [n] having k strong fixed blocks (see first comment for definition).

Original entry on oeis.org

1, 0, 1, 1, 1, 3, 3, 14, 9, 1, 77, 38, 5, 497, 198, 25, 3676, 1229, 134, 1, 30677, 8819, 815, 9, 285335, 71825, 5657, 63, 2928846, 654985, 44549, 419, 1, 32903721, 6615932, 394266, 2868, 13, 401739797, 73357572, 3883182, 20932, 117, 5298600772, 886078937, 42174500, 165662, 928, 1

Offset: 0

Emeric Deutsch, Apr 18 2011

A fixed block of a permutation p is a maximal sequence of consecutive fixed points of p. For example, the permutation 213486759 has 3 fixed blocks: 34, 67, and 9. A fixed block f of a permutation p is said to be strong if all the entries to the left (right) of f are smaller (larger) than all the entries of f. In the above example, only 34 and 9 are strong fixed blocks.
Apparently, row n has 1+ceiling(n/3) entries.
Sum of entries in row n is n!.
T(n,0) = A052186(n).
Sum_{k>=0} k*T(n,k) = A186374(n).
Entries obtained by direct counting (via Maple).
In general, column k > 1 is asymptotic to (k-1) * n! / n^(3*k-4). - Vaclav Kotesovec, Aug 29 2014

			T(3,1) = 3 because we have [123], [1]32, and 21[3] (the strong fixed blocks are shown between square brackets).
T(7,3) = 1 because we have [1]32[4]65[7] (the strong fixed blocks are shown between square brackets).
Triangle starts:
        1;
        0,      1;
        1,      1;
        3,      3;
       14,      9,     1;
       77,     38,     5;
      497,    198,    25;
     3676,   1229,   134,   1;
    30677,   8819,   815,   9;
   285335,  71825,  5657,  63;
  2928846, 654985, 44549, 419,  1;

Alois P. Heinz, Rows n = 0..200, flattened

Cf. A000142, A186374, A052186, A145878.

Columns k=0-10 give: A052186, A225960, A225963, A225964, A225965, A225966, A225967, A225968, A225969, A225970, A225971. - Alois P. Heinz, May 22 2013

b:= proc(n) b(n):=-`if`(n<0, 1, add(b(n-i-1)*i!, i=0..n)) end:
f:= proc(n) f(n):=`if`(n<=0, 0, b(n-1)+f(n-1)) end:
B:= proc(n, k) option remember; `if`(k=0, 0, `if`(k=1, f(n),
      add((f(n-i)-1)*B(i,k-1), i=3*k-5..n-3)))
    end:
T:= proc(n, k) option remember; `if`(k=0, b(n),
      add(b(n-i)*B(i, k), i=3*k-2..n))
    end:
seq(seq(T(n, k), k=0..ceil(n/3)), n=0..20); # Alois P. Heinz, May 23 2013

b[n_] := b[n] = -If[n<0, 1, Sum[b[n-i-1]*i!, {i, 0, n}]]; f[n_] := f[n] = If[n <= 0, 0, b[n-1] + f[n-1]]; B[n_, k_] :=  B[n, k] = If[k == 0, 0, If[k == 1, f[n],  Sum[(f[n-i]-1)*B[i, k-1], {i, 3*k-5, n-3}]]]; T[n_, k_] := T[n, k] = If[k == 0, b[n], Sum[b[n-i]*B[i, k], {i, 3*k-2, n}]]; Table[Table[T[n, k], {k, 0, Ceiling[ n/3]}], {n, 0, 20}] // Flatten (* Jean-François Alcover, Feb 20 2015, after Alois P. Heinz *)

Rows n=11-13 (16 terms) from Alois P. Heinz, May 22 2013

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A260957 Coefficients in asymptotic expansion of sequence A225960.

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A186373 Triangle read by rows: T(n,k) is the number of permutations of [n] having k strong fixed blocks (see first comment for definition).

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