A287256 -id:A287256 - cp's OEIS frontend

A287216 Number A(n,k) of set partitions of [n] such that all absolute differences between least elements of consecutive blocks are <= k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 4, 1, 1, 1, 2, 5, 9, 1, 1, 1, 2, 5, 14, 23, 1, 1, 1, 2, 5, 15, 44, 66, 1, 1, 1, 2, 5, 15, 51, 152, 210, 1, 1, 1, 2, 5, 15, 52, 191, 571, 733, 1, 1, 1, 2, 5, 15, 52, 202, 780, 2317, 2781, 1, 1, 1, 2, 5, 15, 52, 203, 857, 3440, 10096, 11378, 1

Offset: 0

Alois P. Heinz, May 21 2017

			A(4,0) = 1: 1234.
A(4,1) = 9: 1234, 134|2, 13|24, 14|23, 1|234, 14|2|3, 1|24|3, 1|2|34, 1|2|3|4.
A(4,2) = 14: 1234, 124|3, 12|34, 12|3|4, 134|2, 13|24, 13|2|4, 14|23, 1|234, 1|23|4, 14|2|3, 1|24|3, 1|2|34, 1|2|3|4.
A(5,1) = 23: 12345, 1345|2, 134|25, 135|24, 13|245, 145|23, 14|235, 15|234, 1|2345, 145|2|3, 14|25|3, 14|2|35, 15|24|3, 1|245|3, 1|24|35, 15|2|34, 1|25|34, 1|2|345, 15|2|3|4, 1|25|3|4, 1|2|35|4, 1|2|3|45, 1|2|3|4|5.
Square array A(n,k) begins:
  1,   1,   1,   1,   1,   1,   1,   1, ...
  1,   1,   1,   1,   1,   1,   1,   1, ...
  1,   2,   2,   2,   2,   2,   2,   2, ...
  1,   4,   5,   5,   5,   5,   5,   5, ...
  1,   9,  14,  15,  15,  15,  15,  15, ...
  1,  23,  44,  51,  52,  52,  52,  52, ...
  1,  66, 152, 191, 202, 203, 203, 203, ...
  1, 210, 571, 780, 857, 876, 877, 877, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
Wikipedia, Partition of a set

Columns k=0-10 give: A000012, A026898(n-1) for n>0, A287252, A287253, A287254, A287255, A287256, A287257, A287258, A287259, A287260.
Main diagonal gives A000110.
Cf. A287214, A287215, A287417, A287641.

b:= proc(n, k, m, l) option remember; `if`(n<1, 1,
     `if`(l-n>k, 0, b(n-1, k, m+1, n))+m*b(n-1, k, m, l))
    end:
A:= (n, k)-> b(n-1, min(k, n-1), 1, n):
seq(seq(A(n, d-n), n=0..d), d=0..12);

b[n_, k_, m_, l_] := b[n, k, m, l] = If[n < 1, 1, If[l - n > k, 0, b[n - 1, k, m + 1, n]] + m*b[n - 1, k, m, l]];
A[n_, k_] := b[n - 1, Min[k, n - 1], 1, n];
Table[A[n, d - n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Apr 30 2018, after Alois P. Heinz *)

A(n,k) = Sum_{j=0..k} A287215(n,j).

A322879 Number of set partitions of [n] such that the maximal absolute difference between the least elements of consecutive blocks equals six.

Original entry on oeis.org

0, 1, 35, 413, 3709, 30396, 242366, 1934021, 15653524, 129267234, 1091892025, 9444993005, 83722879838, 760771479660, 7087056828919, 67674638461955, 662248199987728, 6638947646238102, 68153572265860585, 716151632403862252, 7699437833837232406

Offset: 6

Alois P. Heinz, Dec 29 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 6..581
Wikipedia, Partition of a set

Column k=6 of A287215.

Cf. A287255, A287256.

b:= proc(n, k, m, l) option remember; `if`(n<1, 1,
     `if`(l-n>k, 0, b(n-1, k, m+1, n))+m*b(n-1, k, m, l))
    end:
A:= (n, k)-> b(n-1, min(k, n-1), 1, n):
a:= n-> (k-> A(n, k)-A(n, k-1))(6):
seq(a(n), n=6..30);

a(n) = A287256(n) - A287255(n).

A322880 Number of set partitions of [n] such that the maximal absolute difference between the least elements of consecutive blocks equals seven.

Original entry on oeis.org

0, 1, 67, 1059, 11373, 106256, 940608, 8193031, 71568443, 633413486, 5710573774, 52579554510, 494986212033, 4766754646529, 46966622143740, 473477764449909, 4883216510830794, 51513192445470426, 555667122424414886, 6127026713823497897, 69034256792655555566

Offset: 7

Alois P. Heinz, Dec 29 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 7..580
Wikipedia, Partition of a set

Column k=7 of A287215.

Cf. A287256, A287257.

b:= proc(n, k, m, l) option remember; `if`(n<1, 1,
     `if`(l-n>k, 0, b(n-1, k, m+1, n))+m*b(n-1, k, m, l))
    end:
A:= (n, k)-> b(n-1, min(k, n-1), 1, n):
a:= n-> (k-> A(n, k)-A(n, k-1))(7):
seq(a(n), n=7..30);

a(n) = A287257(n) - A287256(n).

cp's OEIS Frontend

A287216 Number A(n,k) of set partitions of [n] such that all absolute differences between least elements of consecutive blocks are <= k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A322879 Number of set partitions of [n] such that the maximal absolute difference between the least elements of consecutive blocks equals six.

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Author

Keywords

Links

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Programs

Maple

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A322880 Number of set partitions of [n] such that the maximal absolute difference between the least elements of consecutive blocks equals seven.

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Author

Keywords

Links

Crossrefs

Programs

Maple

Formula