A147679 Triangle read by rows: T(n,k) (n >= 1, 0 <= k <= n-1) is the number of permutations of [0..(n-1)] of spread k.
1, 1, 1, 0, 3, 3, 4, 8, 4, 8, 20, 25, 25, 25, 25, 144, 108, 108, 144, 108, 108, 630, 735, 735, 735, 735, 735, 735, 5696, 4608, 5248, 4608, 5696, 4608, 5248, 4608, 39366, 40824, 40824, 39285, 40824, 40824, 39285, 40824, 40824, 366400, 362000, 362000, 362000, 362000, 366400, 362000, 362000, 362000, 362000
Offset: 1
Examples
Triangle begins:
1
1 1
0 3 3
4 8 4 8
20 25 25 25 25
144 108 108 144 108 108
...
Links
- Seiichi Manyama, Rows n = 1..13, flattened
- R. L. Graham and D. H. Lehmer, On the Permanent of Schur's Matrix, J. Australian Math. Soc., 21A (1976), 487-497.
Crossrefs
Programs
-
Maple
b:= proc(n) option remember; local l, p, r; l:= array([i$i=0..n-1]); r:= array([0$i=1..n]); p:= proc(t,s) local d, h, j; if t=n then d:= ((s+(n-1)*l[n]) mod n) +1; r[d]:= r[d]+1 else for j from t to n do l[t],l[j]:= l[j],l[t]; p(t+1, (s+(t-1)*l[t]) ) od; h:= l[t]; for j from t to n-1 do l[j]:= l[j+1] od; l[n]:= h fi end; p(1,0); eval(r) end: T:= (n,k)-> b(n)[k+1]: seq (seq (T(n,k), k=0..n-1), n=1..10); -
Mathematica
b[n_] := b[n] = Module[{l, p, r}, l = Range[0, n-1]; r = Array[0&, n]; p [t_, s_] := Module[{d, h, j}, If[t == n, d = Mod[s+(n-1)*l[[n]], n]+1; r[[d]] = r[[d]]+1, For[j = t, j <= n, j++, {l[[t]], l[[j]]} = {l[[j]], l[[t]]}; p[t+1, s+(t-1)*l[[t]]]]; h = l[[t]]; For[j = t, j <= n-1, j++, l[[j]] = l[[j+1]]]; l[[n]] = h]]; p[1, 0]; r]; t[n_, k_] := b[n][[k+1]]; Table [Print[t[n, k]]; t[n, k], {n, 1, 10}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, Apr 17 2014, after Alois P. Heinz *) -
Sage
@CachedFunction def A147679_row(n): row = [0]*n for p in Permutations(range(n)): spread = sum(i*px for i,px in enumerate(p)) % n row[spread] += 1 return row A147679 = lambda n,k: A147679_row(n)[k] # D. S. McNeil, Dec 23 2010
Extensions
Edited by Alois P. Heinz, Dec 22 2010
Comments