cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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

Views

Author

Alois P. Heinz, Apr 24 2023

Keywords

Examples

			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.

Links

Crossrefs

Cf. A000110, A007476, A092920, A210540, A362635, A362637, A362893.

Programs

  • Maple
    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);

A362637 Number of partitions of [n] whose blocks are ordered with increasing least elements and where block i (except possibly the last) has size at least i.

Original entry on oeis.org

1, 1, 2, 4, 10, 30, 96, 323, 1184, 4784, 20708, 93073, 431004, 2080610, 10615276, 57291063, 322921896, 1871715144, 11065738360, 66843918825, 415837464280, 2684434034706, 18010208402784, 124877499979859, 886741484322660, 6399683149311272, 46802092819866340
Offset: 0

Views

Author

Alois P. Heinz, Apr 28 2023

Keywords

Examples

			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) = 10: 1234, 123|4, 124|3, 12|34, 134|2, 13|24, 14|23, 1|234, 1|23|4, 1|24|3.
a(5) = 30: 12345, 1234|5, 1235|4, 123|45, 1245|3, 124|35, 125|34, 12|345, 12|34|5, 12|35|4, 1345|2, 134|25, 135|24, 13|245, 13|24|5, 13|25|4, 145|23, 14|235, 14|23|5, 15|234, 1|2345, 1|234|5, 15|23|4, 1|235|4, 1|23|45, 14|25|3, 15|24|3, 1|245|3, 1|24|35, 1|25|34.

Links

Crossrefs

Cf. A000110, A362549, A362635, A362639.

Programs

  • Maple
    b:= proc(n, t) option remember; `if`(n=0, 1, `if`(n<=t, 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);

A362893 Number of partitions of [n] whose blocks can be ordered such that the i-th block 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, 1, 2, 5, 12, 28, 69, 193, 614, 2103, 7359, 25660, 88914, 309502, 1102146, 4092840, 16046224, 66410789, 286905421, 1273646720, 5729762139, 25881820352, 116872997038, 527375160184, 2384407416357, 10856086444051, 50097994816979, 235937202788389
Offset: 0

Views

Author

Alois P. Heinz, May 08 2023

Keywords

Examples

			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) = 28: 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.
a(7) = 69: 1234567, 12345|67, 12346|57, 12347|56, 1234|567, 12356|47, 12357|46, 1235|467, 12367|45, 1236|457, 1237|456, 123|4567, 12456|37, 12457|36, 1245|367, 12467|35, 1246|357, 1247|356, 124|3567, 12567|34, 1256|347, 1257|346, 125|3467, 1267|345, 126|3457, 127|3456, 12|34567, 12|34|567, 13456|27, 13457|26, 1345|267, 13467|25, 1346|257, 1347|256, 134|2567, 13567|24, 1356|247, 1357|246, 135|2467, 1367|245, 136|2457, 137|2456, 13|24567, 13|24|567, 14567|23, 1456|237, 1457|236, 145|2367, 1467|235, 146|2357, 147|2356, 14|23567, 14|23|567, 1567|234, 156|2347, 157|2346, 15|23467, 167|2345, 16|23457, 17|23456, 1|234567, 1|234|567, 15|23|467, 1|235|467, 16|23|457, 1|236|457, 17|23|456, 1|237|456, 1|23|4567.

Links

Crossrefs

Cf. A000110, A362549, A362635.

Programs

  • Maple
    b:= proc(n, t) option remember; `if`(n=0 or n=t, 1,
      add(b(n-j, t+1)*binomial(n-t, j-t), j=t..n))
    end:
a:= n-> b(n, 1):
seq(a(n), n=0..28);

A362638 Number of partitions of [n] whose blocks are ordered with increasing least elements and where block i has size at most i.

Original entry on oeis.org

1, 1, 1, 2, 4, 11, 34, 117, 458, 1960, 9113, 45730, 246220, 1411820, 8584314, 55121249, 372445462, 2639750490, 19570232608, 151390999949, 1219308754065, 10204045406690, 88571532840007, 796100505025761, 7398510884959638, 70995106473861076, 702542384597885615
Offset: 0

Views

Author

Alois P. Heinz, Apr 28 2023

Keywords

Examples

			a(0) = 1: ().
a(1) = 1: 1.
a(2) = 1: 1|2.
a(3) = 2: 1|23, 1|2|3.
a(4) = 4: 1|23|4, 1|24|3, 1|2|34, 1|2|3|4.
a(5) = 11: 1|23|45, 1|23|4|5, 1|24|35, 1|24|3|5, 1|25|34, 1|2|345, 1|2|34|5, 1|25|3|4, 1|2|35|4, 1|2|3|45, 1|2|3|4|5.
a(6) = 34: 1|23|456, 1|23|45|6, 1|23|46|5, 1|23|4|56, 1|23|4|5|6, 1|24|356, 1|24|35|6, 1|24|36|5, 1|24|3|56, 1|24|3|5|6, 1|25|346, 1|25|34|6, 1|26|345, 1|2|345|6, 1|26|34|5, 1|2|346|5, 1|2|34|56, 1|2|34|5|6, 1|25|36|4, 1|25|3|46, 1|25|3|4|6, 1|26|35|4, 1|2|356|4, 1|2|35|46, 1|2|35|4|6, 1|26|3|45, 1|2|36|45, 1|2|3|456, 1|2|3|45|6, 1|26|3|4|5, 1|2|36|4|5, 1|2|3|46|5, 1|2|3|4|56, 1|2|3|4|5|6.

Links

Crossrefs

Cf. A000110, A362635.

Programs

  • Maple
    b:= proc(n, t) option remember; `if`(n=0, 1, add(
      b(n-j, t+1)*binomial(n-1, j-1), j=1..min(t, n)))
    end:
a:= n-> b(n, 1):
seq(a(n), n=0..30);
Showing 1-4 of 4 results.

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