1,5

Row sums are the factorial numbers (A000142). T(n,1)=1 (the identity permutation). T(n,2)=A065608(n)=(sum of divisors of n)-(number of divisors of n). T(n,n)=A099152(n). In the Maple program define n (<=10) to obtain row n.

T(n,k) is also the number of permutations p of {1,2,...,n} such that the set {p(i) + i, i=1,2,...,n} has exactly k elements (1<=k<=n). Example: T(4,2)=4 because we have 1432, 3412, 2143 and 3214. [From Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 28 2008]

M. Alekseev, E. Deutsch, and J. H. Steelman, Problem 11281, Amer. Math. Monthly, 116, No. 5, 2009, p. 465. [From Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 23 2009]

T(4,2)=4 because we have 4123, 3412, 2143 and 2341.

Triangle starts:

1;

1,1;

1,2,3;

1,4,12,7;

1,4,38,54,23;

n:=7: with(combinat): P:=permute(n): for j from 1 to n! do c[j]:=0 od: for j from 1 to n! do if nops({seq(P[j][i]-i, i=1..n)}) = 1 then c[1]:=c[1]+1 elif nops({seq(P[j][i]-i, i=1..n)}) = 2 then c[2]:=c[2]+1 elif nops({seq(P[j][i]-i, i=1..n)}) = 3 then c[3]:=c[3]+1 elif nops({seq(P[j][i]-i, i=1..n)}) = 4 then c[4]:=c[4]+1 elif nops({seq(P[j][i]-i, i=1..n)}) = 5 then c[5]:=c[5]+1 elif nops({seq(P[j][i]-i, i=1..n)}) = 6 then c[6]:=c[6]+1 elif nops({seq(P[j][i]-i, i=1..n)}) = 7 then c[7]:=c[7]+1 elif nops({seq(P[j][i]-i, i=1..n)}) = 8 then c[8]:=c[8]+1 elif nops({seq(P[j][i]-i, i=1..n)}) = 9 then c[9]:=c[9]+1 elif nops({seq(P[j][i]-i, i=1..n)}) = 10 then c[10]:=c[10]+1 else fi od: seq(c[i], i=1..n);

Cf. A000142, A065608, A099152.

Sequence in context: A078990 A079639 A104694 this_sequence A080063 A140706 A139764

Adjacent sequences: A125179 A125180 A125181 this_sequence A125183 A125184 A125185

nonn,tabl

Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 24 2006