|
| |
|
|
A125182
|
|
Triangle read by rows: T(n,k) is 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).
|
|
1
| |
|
|
1, 1, 1, 1, 2, 3, 1, 4, 12, 7, 1, 4, 38, 54, 23, 1, 8, 77, 248, 303, 83, 1, 6, 160, 824, 2008, 1636, 405, 1, 11, 285, 2320, 9449, 15789, 10352, 2113, 1, 10, 476, 5564, 37237, 102726, 133293, 70916, 12657
(list; table; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 1,5
|
|
|
COMMENTS
| 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]
|
|
|
REFERENCES
| 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]
|
|
|
EXAMPLE
| 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;
|
|
|
MAPLE
| 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);
|
|
|
CROSSREFS
| Cf. A000142, A065608, A099152.
Sequence in context: A176566 A079639 A104694 * A169625 A201737 A080063
Adjacent sequences: A125179 A125180 A125181 * A125183 A125184 A125185
|
|
|
KEYWORD
| nonn,tabl
|
|
|
AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 24 2006
|
| |
|
|
