A287259 -id:A287259 - 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).

A322882 Number of set partitions of [n] such that the maximal absolute difference between the least elements of consecutive blocks equals nine.

Original entry on oeis.org

0, 1, 259, 7851, 123693, 1517480, 16628928, 172861375, 1757583339, 17780116911, 180778826049, 1858914009077, 19407229306905, 206203531592425, 2232778235440364, 24655217395787251, 277719538910592762, 3191229583066629810, 37404691679158439649

Offset: 9

Alois P. Heinz, Dec 29 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 9..579
Wikipedia, Partition of a set

Column k=9 of A287215.

Cf. A287258, A287259.

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))(9):
seq(a(n), n=9..30);

a(n) = A287259(n) - A287258(n).

A322883 Number of set partitions of [n] such that the maximal absolute difference between the least elements of consecutive blocks equals ten.

Original entry on oeis.org

0, 1, 515, 22253, 430909, 6094476, 74507486, 847129333, 9296465127, 100540964675, 1085004090887, 11775039127122, 129155075413877, 1436488582202316, 16235344928131625, 186710546094489052, 2186538096666720967, 26085011069325363939, 317049671003606985326

Offset: 10

Alois P. Heinz, Dec 29 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 10..579
Wikipedia, Partition of a set

Column k=10 of A287215.

Cf. A287259, A287260.

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))(10):
seq(a(n), n=10..30);

a(n) = A287260(n) - A287259(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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A322882 Number of set partitions of [n] such that the maximal absolute difference between the least elements of consecutive blocks equals nine.

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Author

Keywords

Links

Crossrefs

Programs

Maple

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A322883 Number of set partitions of [n] such that the maximal absolute difference between the least elements of consecutive blocks equals ten.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Formula