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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). 2
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, 1, 14, 799, 13172, 122708, 536900, 1158368, 1177168, 537373, 82297 (list; table; graph; refs; listen; history; text; 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 first 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. - Emeric Deutsch, Nov 28 2008

REFERENCES

M. Alekseev, E. Deutsch, and J. H. Steelman, Problem 11281, Amer. Math. Monthly, 116, No. 5, 2009, p. 465. - Emeric Deutsch, Apr 23 2009

LINKS

Alois P. Heinz, Rows n = 1..12, flattened

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);

# second Maple program:

b:= proc(i, p, s) option remember; `if`(p={}, x^nops(s),

      add(b(i+1, p minus {t}, s union {t+i}), t=p))

    end:

T:= n->(p->seq(coeff(p, x, i), i=1..n))(b(1, {$1..n}, {})):

seq(T(n), n=1..9);  # Alois P. Heinz, May 04 2014

MATHEMATICA

b[i_, p_List, s_List] := b[i, p, s] = If[p == {}, x^Length[s], Sum[b[i+1, p ~Complement~ {t}, s ~Union~ {t+i}], {t, p}]]; T[n_] := Function[{p}, Table[ Coefficient[p, x, i], {i, 1, n}]][b[1, Range[n], {}]]; Table[T[n], {n, 1, 9}] // Flatten (* Jean-Fran├žois Alcover, Feb 19 2015, after Alois P. Heinz *)

CROSSREFS

Cf. A000142, A065608, A099152.

Sequence in context: A176566 A079639 A104694 * A270312 A169625 A264794

Adjacent sequences:  A125179 A125180 A125181 * A125183 A125184 A125185

KEYWORD

nonn,tabl

AUTHOR

Emeric Deutsch, Nov 24 2006

STATUS

approved

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Last modified February 19 18:31 EST 2018. Contains 299356 sequences. (Running on oeis4.)