|
| |
|
|
A105552
|
|
Irregular triangle T(n,k) read down columns: the number of compositions c of n with largest_part(c)+length(c)=k+1 in row n, column k.
|
|
5
| |
|
|
1, 2, 4, 1, 7, 5, 2, 11, 14, 12, 5, 1, 16, 30, 39, 32, 18, 7, 2, 22, 55, 95, 113, 101, 71, 41, 18, 6, 1, 29, 91, 195, 299, 357, 350, 292, 207, 126, 64, 27, 9, 2, 37, 140, 357, 664, 978, 1204, 1283, 1198, 992, 731, 482, 284, 148, 66, 25, 7, 1, 46, 204, 602, 1309, 2274, 3329, 4253
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 1,2
|
|
|
COMMENTS
| For each of the A000041(n) partitions of n, one can assign a weight to the partition which counts the permutations of that partition,
given by the multinomial coefficient derived from the frequency representation of the parts.
An equivalent representation is given by writing down all compositions of n.
The entries count those partitions multiplied by their weights (= compositions) of n where the sum of the largest addend
plus number of parts equals k+1. Only nonzero counts are entered into the sequence.
Each entry can also be interpreted as counting a subset of numbers in A055932, because there is
a 1-to-1 correspondence between their prime signature and ordered partitions.
Each diagonal of T(n,k) can be decomposed into p(n) sequences. For example,
A086602 = 2 12 39 95 195 ... is the sum of
A000330 = 1 5 14 30 55 ... plus
A001296 = 1 7 25 65 140 ...
The main diagonal and subdiagonals in order of appearance are A000124, A000330, A086602, A089574, A107600, A107601, A109125,...
|
|
|
FORMULA
| Row sums: sum_{k=0..n} T(n,k) = 2^(n-1).
Column sums: sum_{n=k..infinity} T(n,k) = A047970(n).
|
|
|
EXAMPLE
| The row n=7 starts from the partitions (weights in parentheses) 7 (1),
6+1 (2), 5+2 (2), 4+3 (2), 5+1+1 (3), 4+2+1 (6=3!/1!/1!/1!), 3+3+1 (3), 3+2+2 (3),
4+1+1+1 (4=4!/1!/3!), 3+2+1+1 (12 = 4!/1!/1!/2!), 2+2+2+1 (4), 3+1+1+1+1+1 (5),
2+2+1+1+1 (10=5!/2!/3!), 2+1+1+1+1 (6), 1+1+1+1+1+1 (1). Then T(7,7)= 1+2+3+4+5+6+1 = 22 is the
sum of the weights of partitions with largest part 7 and length 1,
largest part 6 and length 2,... largest part 1 and length 7.
T(7,6) = 2+6+12+10 = 30 is the sum of the weights of the partitions with largest part 6 and length 1,
largest part 5 and length 2, ..., largest part 1 and length 6.
T(7,5) = 2+3+3+4 = 12 collects all the partitions with largest part 5 and length 1 down to
largest part 1 and length 5.
The array has A033638(k) nonzero entries per column, starting at n=1 as :
1
..2
....4
....1..7
.......5..11
.......2..14..16
..........12..30..22
...........5..39..55..29
...........1..32..95..91..37
..............18.113.195.140
...............7.101.299.357
...............2
|
|
|
MAPLE
| A033638 := proc(n) ( (7+(-1)^n)/2 + n^2 )/4 ; end proc:
freq := proc(L, n) local a, p; a := 0 ; for p in L do if p = n then a := a+1 ; end if; end do: a ; end proc:
M3 := proc(L) local a, i; a := factorial(nops(L)) ; for i in convert(L, set) do a := a/factorial(freq(L, i)) ; end do: a ; end proc:
A105552 := proc(n, k) local p, a, l ; a := 0 ; for p in combinat[partition](n) do if max(op(p)) + nops(p) = k+1 then a := a+ M3(p); end if; end do ; a ; end proc:
for k from 1 to 15 do for n from k to k+A033638(k)+1 do T := A105552(n, k) ; if T >0 then printf("%d, ", A105552(n, k)) ; end if; end do: printf("\n") ; end do: # R. J. Mathar, Jun 26 2010
|
|
|
CROSSREFS
| Cf. A047969, A047970, A055932, A057335, A083480, A083906, A089349, A033638.
Sequence in context: A142146 A143350 A119303 * A112852 A121531 A127554
Adjacent sequences: A105549 A105550 A105551 * A105553 A105554 A105555
|
|
|
KEYWORD
| nonn,tabf
|
|
|
AUTHOR
| Alford Arnold (Alford1940(AT)aol.com), May 03 2005
|
|
|
EXTENSIONS
| Definition clarified by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jun 26 2010
|
| |
|
|
