A124773
Number of permutations associated with compositions in standard order.
Original entry on oeis.org
1, 1, 1, 1, 2, 2, 1, 1, 6, 6, 3, 3, 2, 2, 1, 1, 24, 24, 12, 12, 8, 8, 4, 4, 6, 6, 3, 3, 2, 2, 1, 1, 120, 120, 60, 60, 40, 40, 20, 20, 30, 30, 15, 15, 10, 10, 5, 5, 24, 24, 12, 12, 8, 8, 4, 4, 6, 6, 3, 3, 2, 2, 1, 1, 720, 720, 360, 360, 240, 240, 120, 120, 180, 180, 90, 90, 60, 60, 30
Offset: 0
Composition number 11 is 2,1,1; the associated permutations are (12)(3)(4), (13)(2)(4) and (14)(2)(3), so a(11) = 3.
The table starts:
1
1
1 1
2 2 1 1
A371417
Triangle read by rows: T(n,k) is the number of complete compositions of n with k parts.
Original entry on oeis.org
1, 0, 1, 0, 0, 1, 0, 0, 2, 1, 0, 0, 0, 3, 1, 0, 0, 0, 3, 4, 1, 0, 0, 0, 6, 6, 5, 1, 0, 0, 0, 0, 16, 10, 6, 1, 0, 0, 0, 0, 12, 30, 15, 7, 1, 0, 0, 0, 0, 12, 35, 50, 21, 8, 1, 0, 0, 0, 0, 24, 50, 75, 77, 28, 9, 1, 0, 0, 0, 0, 0, 90, 126, 140, 112, 36, 10, 1
Offset: 0
The triangle begins:
k=0 1 2 3 4 5 6 7 8 9 10
n=0: 1;
n=1: 0, 1;
n=2: 0, 0, 1;
n=3: 0, 0, 2, 1;
n=4: 0, 0, 0, 3, 1;
n=5: 0, 0, 0, 3, 4, 1;
n=6: 0, 0, 0, 6, 6, 5, 1;
n=7: 0, 0, 0, 0, 16, 10, 6, 1;
n=8: 0, 0, 0, 0, 12, 30, 15, 7, 1;
n=9: 0, 0, 0, 0, 12, 35, 50, 21, 8, 1;
n=10: 0, 0, 0, 0, 24, 50, 75, 77, 28, 9, 1;
...
For n = 5 there are a total of 8 complete compositions:
T(5,3) = 3: (221), (212), (122)
T(5,4) = 4: (2111), (1211), (1121), (1112)
T(5,5) = 1: (11111)
A107428 counts gap-free compositions.
A251729 counts gap-free but not complete compositions.
Cf.
A107429 (row sums give complete compositions of n),
A000670 (column sums),
A152947 (number of nonzero terms per column).
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b:= proc(n, i, t) option remember; `if`(n=0,
`if`(i=0, t!, 0), `if`(i<1 or n (p-> seq(coeff(p, x, i), i=0..n))(add(b(n, i, 0), i=0..n)):
seq(T(n), n=0..12); # Alois P. Heinz, Apr 03 2024
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G(N)={ my(z='z+O('z^N)); Vec(sum(i=1,N,z^(i*(i+1)/2)*t^i*prod(j=1,i,sum(k=0,N, (z^(j*k)*t^k)/(k+1)!))))}
my(v=G(10)); for(n=0, #v, if(n<1,print([1]), my(p=v[n], r=vector(n+1)); for(k=0, n, r[k+1] =k!*polcoeff(p, k)); print(r)))
Showing 1-2 of 2 results.
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