0,7

Row sums are the factorials (A000142).

T(n,0)=A0521861(n).

Sum(T(n,k),k=1..n)=A006932(n).

Sum(k*T(n,k),k=0..n)=A003149(n-1).

Conjectures: T(3+k,k)=3k+3, T(4+k,k)=(k+1)(k+28)/2, T(5+k,k)=(k+1)(3k+77), T(6+k,k)=(k+1)(k^2+110k+2982)/6, T(7+k,k)=(k+1)(3k^2+235k+7352)/2.

The Maple program, in the present form, yields row 7 of the triangle. Row n (0<=n<=10) is obtained by changing the value of n at the beginning of the program.

Stanley, R. P., Enumerative Combinatorics, Volume 1 (1986), p. 49.

Problem E3467, Amer. Math. Monthly, 100 (1993), 800-801.

V. Strehl, The average number of splitters in a random permutation [Unpublished; included here with the author's permission.]

T(5,3)=4 because we have 1'2'3'54, 1'2'435', 1'324'5' and 213'4'5' (the strong fixed points are marked).

Triangle starts:

1;

0,1;

1,0,1;

3,2,0,1;

14,6,3,0,1;

77,29,9,4,0,1;

n:=7: sfix:=proc(p) local ct, i: ct:= 0: for i to nops(p) do if p[i]=i and `subset`({seq(p[j], j=1..i-1)}, {seq(k, k=1..i-1)})=true then ct:=ct+1 else end if end do: ct end proc: with(combinat): P:=permute(n): s:=[seq(sfix(P[j]), j= 1..factorial(n))]: for i from 0 to n do a[i]:=0 end do: for j to factorial(n) do if s[j]=0 then a[0]:=a[0]+1 elif s[j]=1 then a[1]:=a[1]+1 elif s[j]=2 then a[2]:=a[2]+1 elif s[j]=3 then a[3]:=a[3]+1 elif s[j]=4 then a[4]:=a[4]+1 elif s[j]=5 then a[5]:=a[5]+1 elif s[j]=6 then a[6]:=a[6]+1 elif s[j]=7 then a[7]:= a[7]+1 elif s[j]=8 then a[8]:=a[8]+1 elif s[j]=9 then a[9]:=a[9]+1 elif s[j]= 10 then a[10]:=a[10]+1 end if end do: seq(a[k], k=0..n); # yields row m of the triangle, where m is the value of n specified at the beginning of the program

Cf. A000142, A052861, A006932, A003149

Sequence in context: A154477 A142071 A118972 this_sequence A112606 A108512 A054503

Adjacent sequences: A145875 A145876 A145877 this_sequence A145879 A145880 A145881

nonn,tabl

Emeric Deutsch (deutsch(AT)duke.poly.edu), Oct 29 2008