A287254 -id:A287254 - 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).

A322877 Number of set partitions of [n] such that the maximal absolute difference between the least elements of consecutive blocks equals four.

Original entry on oeis.org

0, 1, 11, 77, 493, 3124, 20019, 130916, 878249, 6063134, 43144661, 316670184, 2397764986, 18726889938, 150814853887, 1251834352246, 10703915163764, 94227518620167, 853463133257984, 7948557602950239, 76069254546156710, 747596311576859585, 7540213445348427312

Offset: 4

Alois P. Heinz, Dec 29 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 4..584
Wikipedia, Partition of a set

Column k=4 of A287215.

Cf. A287253, A287254.

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))(4):
seq(a(n), n=4..30);

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];
a[n_] := With[{k = 4}, A[n, k] - A[n, k - 1]];
a /@ Range[4, 30] (* Jean-François Alcover, May 05 2020, after Maple *)

a(n) = A287254(n) - A287253(n).

A322878 Number of set partitions of [n] such that the maximal absolute difference between the least elements of consecutive blocks equals five.

Original entry on oeis.org

0, 1, 19, 171, 1293, 9320, 66992, 488526, 3637440, 27735903, 216992278, 1743777862, 14401360577, 122242150172, 1066284279026, 9554869690126, 87923414758506, 830459368379431, 8047463255217118, 79967170844047637, 814439368083686232, 8497321384591725159

Offset: 5

Alois P. Heinz, Dec 29 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 5..582
Wikipedia, Partition of a set

Column k=5 of A287215.

Cf. A287254, A287255.

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))(5):
seq(a(n), n=5..30);

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];
a[n_] := With[{k = 5}, A[n, k] - A[n, k - 1]];
a /@ Range[5, 30] (* Jean-François Alcover, May 05 2020, after Maple *)

a(n) = A287255(n) - A287254(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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A322877 Number of set partitions of [n] such that the maximal absolute difference between the least elements of consecutive blocks equals four.

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Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

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A322878 Number of set partitions of [n] such that the maximal absolute difference between the least elements of consecutive blocks equals five.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula