A272498
Number of ordered set partitions of [n] with nondecreasing block sizes and maximal block size equal to eight.
Original entry on oeis.org
1, 9, 135, 1650, 23265, 316602, 4810806, 73880235, 1229123610, 21174714990, 388551217626, 7431930745668, 150102842702670, 3162843042018660, 69923464752835980, 1611044465380180974, 38759812951913315262, 969843174518264324850, 25246982138722170061950
Offset: 8
-
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
b(n, i-1)+`if`(i>n, 0, binomial(n, i)*b(n-i, i))))
end:
a:= n-> (k-> b(n, k) -b(n, k-1))(8):
seq(a(n), n=8..30);
A272499
Number of ordered set partitions of [n] with nondecreasing block sizes and maximal block size equal to nine.
Original entry on oeis.org
1, 10, 165, 2200, 33605, 492492, 8018010, 131342640, 2321677930, 42349478600, 820275716546, 16515429370440, 350240612952230, 7731410818511380, 178693701272340540, 4296129057927296304, 107666415418378051950, 2801776425029317564400, 75741144900761549630850
Offset: 9
-
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
b(n, i-1)+`if`(i>n, 0, binomial(n, i)*b(n-i, i))))
end:
a:= n-> (k-> b(n, k) -b(n, k-1))(9):
seq(a(n), n=9..30);
A272500
Number of ordered set partitions of [n] with nondecreasing block sizes and maximal block size equal to ten.
Original entry on oeis.org
1, 11, 198, 2860, 47047, 738738, 12828816, 223282488, 4179020274, 80464009340, 1640551617848, 34682405557800, 770529476530814, 17782248154604934, 428864975324828328, 10740325059575465640, 279932748231053890830, 7564798231253861700960, 212075260563875086898520
Offset: 10
-
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
b(n, i-1)+`if`(i>n, 0, binomial(n, i)*b(n-i, i))))
end:
a:= n-> (k-> b(n, k) -b(n, k-1))(10):
seq(a(n), n=10..30);
A339030
T(n, k) = Sum_{p in P(n, k)} card(p), where P(n, k) is the set of set partitions of {1,2,...,n} where the largest block has size k and card(p) is the number of blocks of p. Triangle T(n, k) for 0 <= k <= n, read by rows.
Original entry on oeis.org
1, 0, 1, 0, 2, 1, 0, 3, 6, 1, 0, 4, 24, 8, 1, 0, 5, 85, 50, 10, 1, 0, 6, 300, 280, 75, 12, 1, 0, 7, 1071, 1540, 525, 105, 14, 1, 0, 8, 3976, 8456, 3570, 840, 140, 16, 1, 0, 9, 15219, 47208, 24381, 6552, 1260, 180, 18, 1
Offset: 0
Triangle starts:
0: [1]
1: [0, 1]
2: [0, 2, 1]
3: [0, 3, 6, 1]
4: [0, 4, 24, 8, 1]
5: [0, 5, 85, 50, 10, 1]
6: [0, 6, 300, 280, 75, 12, 1]
7: [0, 7, 1071, 1540, 525, 105, 14, 1]
8: [0, 8, 3976, 8456, 3570, 840, 140, 16, 1]
9: [0, 9, 15219, 47208, 24381, 6552, 1260, 180, 18, 1]
.
T(4,0) = 0 = 0*card({})
T(4,1) = 4 = 4*card({1|2|3|4}).
T(4,2) = 24 = 3*card({12|3|4, 13|2|4, 1|23|4, 14|2|3, 1|24|3, 1|2|34})
+ 2*card({12|34, 13|24, 14|23}).
T(4,3) = 8 = 2*card({123|4, 124|3, 134|2, 1|234}).
T(4,4) = 1 = 1*card({1234}).
.
Seen as the projection of a 2-dimensional statistic this is, for n = 6:
[ 0 0 0 0 0 0 0]
[ 0 0 0 0 0 0 6]
[ 0 0 0 45 180 75 0]
[ 0 0 20 180 80 0 0]
[ 0 0 30 45 0 0 0]
[ 0 0 12 0 0 0 0]
[ 0 1 0 0 0 0 0]
The row sum projection gives row 6 of this triangle, and the column sum projection gives [0, 1, 62, 270, 260, 75, 6], which appears in a decapitated version as row 5 in A321331.
Cf.
A005493 with 1 prepended are the row sums.
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def A339030Row(n):
if n == 0: return [1]
M = matrix(n + 1)
for k in (1..n):
for p in SetPartitions(n):
if p.max_block_size() == k:
M[k, len(p)] += p.cardinality()
return [sum(M[k, j] for j in (0..n)) for k in (0..n)]
for n in (0..9): print(A339030Row(n))