A362635
Number of partitions of [n] whose blocks are ordered with increasing least elements and where block i has size at least i.
Original entry on oeis.org
1, 1, 1, 2, 5, 12, 31, 97, 351, 1318, 4963, 19391, 82531, 386704, 1926907, 9811733, 50252175, 261462430, 1415025895, 8118274255, 49355434511, 312266428040, 2012834117143, 13055850467371, 85215848844559, 565353777291346, 3866868949795579, 27548261709035105
Offset: 0
a(0) = 1: (), the empty partition.
a(1) = 1: 1.
a(2) = 1: 12.
a(3) = 2: 123, 1|23.
a(4) = 5: 1234, 12|34, 13|24, 14|23, 1|234.
a(5) = 12: 12345, 123|45, 124|35, 125|34, 12|345, 134|25, 135|24, 13|245, 145|23, 14|235, 15|234, 1|2345.
a(6) = 31: 123456, 1234|56, 1235|46, 1236|45, 123|456, 1245|36, 1246|35, 124|356, 1256|34, 125|346, 126|345, 12|3456, 1345|26, 1346|25, 134|256, 1356|24, 135|246, 136|245, 13|2456, 1456|23, 145|236, 146|235, 14|2356, 156|234, 15|2346, 16|2345, 1|23456, 1|23|456, 1|24|356, 1|25|346, 1|26|345.
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b:= proc(n, t) option remember; `if`(n=0, 1, add(
b(n-j, t+1)*binomial(n-1, j-1), j=t..n))
end:
a:= n-> b(n, 1):
seq(a(n), n=0..30);
A362549
Number of partitions of [n] whose blocks can be ordered such that the i-th block (except possibly the last) has at least i elements and no block j > i has an element smaller than the i-th smallest element of block i.
Original entry on oeis.org
1, 1, 2, 4, 9, 23, 64, 187, 566, 1777, 5820, 19944, 71343, 264719, 1011292, 3953381, 15756609, 63945484, 264384828, 1115246518, 4806957739, 21189601861, 95516470253, 439777682222, 2064164172616, 9853934668051, 47736608806520, 234235866539512, 1162618720397931
Offset: 0
a(0) = 1: (), the empty partition.
a(1) = 1: 1.
a(2) = 2: 12, 1|2.
a(3) = 4: 123, 12|3, 13|2, 1|23.
a(4) = 9: 1234, 123|4, 124|3, 12|34, 134|2, 13|24, 14|23, 1|234, 1|23|4.
a(5) = 23: 12345, 1234|5, 1235|4, 123|45, 1245|3, 124|35, 125|34, 12|345, 12|34|5, 1345|2, 134|25, 135|24, 13|245, 13|24|5, 145|23, 14|235, 14|23|5, 15|234, 1|2345, 1|234|5, 15|23|4, 1|235|4, 1|23|45.
a(6) = 64: 123456, 12345|6, 12346|5, 1234|56, 12356|4, ..., 1|2356|4, 1|235|46, 16|23|45, 1|236|45, 1|23|456.
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b:= proc(n, t) option remember; `if`(n<=t, 1,
add(b(j, t+1)*binomial(n-t, j), j=0..n-t))
end:
a:= n-> b(n, 1):
seq(a(n), n=0..30);
A362639
Number of partitions of [n] whose blocks are ordered with increasing least elements and where block i (except possibly the last) has size i.
Original entry on oeis.org
1, 1, 1, 1, 2, 3, 4, 15, 36, 70, 120, 756, 2800, 7920, 18900, 40040, 388080, 2106000, 8408400, 27489000, 77837760, 197520960, 2756754000, 20903929200, 113809696000, 497097881280, 1847907341280, 6062876820000, 17990209036800, 343877493960000, 3501594297801600
Offset: 0
a(0) = 1: (), the empty partition.
a(1) = 1: 1.
a(2) = 1: 1|2.
a(3) = 1: 1|23.
a(4) = 2: 1|23|4, 1|24|3.
a(5) = 3: 1|23|45, 1|24|35, 1|25|34.
a(6) = 4: 1|23|456, 1|24|356, 1|25|346, 1|26|345.
a(7) = 15: 1|23|456|7, 1|23|457|6, 1|23|467|5, 1|24|356|7, 1|24|357|6, 1|24|367|5, 1|25|346|7, 1|25|347|6, 1|26|345|7, 1|27|345|6, 1|26|347|5, 1|27|346|5, 1|25|367|4, 1|26|357|4, 1|27|356|4.
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b:= proc(n, t) option remember; `if`(n<=t, 1,
b(n-t, t+1)*binomial(n-1, t-1))
end:
a:= n-> b(n, 1):
seq(a(n), n=0..30);
Showing 1-3 of 3 results.