|
|
A080510
|
|
Triangle read by rows: T(n,k) gives the number of set partitions of {1,...,n} with maximum block length k.
|
|
33
|
|
|
1, 1, 1, 1, 3, 1, 1, 9, 4, 1, 1, 25, 20, 5, 1, 1, 75, 90, 30, 6, 1, 1, 231, 420, 175, 42, 7, 1, 1, 763, 2016, 1015, 280, 56, 8, 1, 1, 2619, 10024, 6111, 1890, 420, 72, 9, 1, 1, 9495, 51640, 38010, 12978, 3150, 600, 90, 10, 1, 1, 35695, 276980, 244035, 91938, 24024, 4950, 825, 110, 11, 1
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,5
|
|
COMMENTS
|
Row sums are A000110 (Bell numbers). Second column is A001189 (Degree n permutations of order exactly 2).
Partition product of Product_{j=0..n-1} ((k + 1)*j - 1) and n! at k = -1, summed over parts with equal biggest part (see the Luschny link).
Underlying partition triangle is A036040.
Same partition product with length statistic is A008277.
Construct an array in which the n-th row is the partition function G(n,k), where G(n,1),...,G(n,6) = A000012, A000085, A001680, A001681, A110038, A148092, with the first few rows
1, 2, 4, 10, 26, 76, 232, ... = A000085
1, 2, 5, 14, 46, 166, 652, ... = A001680
1, 2, 5, 15, 51, 196, 827, ... = A001681
1, 2 5 15 52 202 869, ... = A110038
1, 2, 5 15 52 203 876, ... = A148092
...
Rows tend to A000110, the Bell numbers. Taking finite differences from the top, then reorienting, we obtain triangle A080510.
The n-th row of the array is the eigensequence of an infinite lower triangular matrix with n diagonals of Pascal's triangle starting from the right and the rest zeros. (End)
|
|
LINKS
|
|
|
FORMULA
|
E.g.f. for k-th column: exp(exp(x)*GAMMA(k, x)/(k-1)!-1)*(exp(x^k/k!)-1). - Vladeta Jovovic, Feb 04 2005
T(n,0) = [n = 0] (Iverson notation) and for n > 0 and 1 <= m <= n.
T(n,m) = Sum_{a} M(a)|f^a| where a = a_1,...,a_n such that
1*a_1 + 2*a_2 + ... + n*a_n = n and max{a_i} = m, M(a) = n!/(a_1!*...*a_n!),
f^a = (f_1/1!)^a_1*...*(f_n/n!)^a_n and f_n = Product_{j=0..n-1} (-1) = (-1)^n. (End)
T(2n,n) = C(2n,n)*(A000110(n)-1/2) for n>0.
T(n,m) = C(n,m)*A000110(n-m) for 2m > n > 0. (End)
|
|
EXAMPLE
|
T(4,3) = 4 since there are 4 set partitions with longest block of length 3: {{1},{2,3,4}}, {{1,3,4},{2}}, {{1,2,3},{4}} and {{1,2,4},{3}}.
Triangle begins:
1;
1, 1;
1, 3, 1;
1, 9, 4, 1;
1, 25, 20, 5, 1;
1, 75, 90, 30, 6, 1;
1, 231, 420, 175, 42, 7, 1;
1, 763, 2016, 1015, 280, 56, 8, 1;
1, 2619, 10024, 6111, 1890, 420, 72, 9, 1;
...
|
|
MAPLE
|
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(b(n-i*j, i-1) *n!/i!^j/(n-i*j)!/j!, j=0..n/i)))
end:
T:= (n, k)-> b(n, k) -b(n, k-1):
|
|
MATHEMATICA
|
<< DiscreteMath`NewCombinatorica`; Table[Length/@Split[Sort[Max[Length/@# ]&/@SetPartitions[n]]], {n, 12}]
(* Second program: *)
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[b[n-i*j, i-1]*n!/i!^j/(n-i*j)!/j!, {j, 0, n/i}]]]; T[n_, k_] := b[n, k]-b[n, k-1]; Table[Table[T[n, k], {k, 1, n}], {n, 1, 12}] // Flatten (* Jean-François Alcover, Feb 25 2014, after Alois P. Heinz *)
|
|
CROSSREFS
|
Cf. A157396, A157397, A157398, A157399, A157400, A157401, A157402, A157403, A157404, A157405. - Peter Luschny, Mar 09 2009
Columns k=1..10 give: A000012 (for n>0), A001189, A229245, A229246, A229247, A229248, A229249, A229250, A229251, A229252. - Alois P. Heinz, Sep 17 2013
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|