A287641 -id:A287641 - 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).

A287214 Number A(n,k) of set partitions of [n] such that for each block all absolute differences between consecutive elements are <= k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 4, 1, 1, 1, 2, 5, 8, 1, 1, 1, 2, 5, 13, 16, 1, 1, 1, 2, 5, 15, 34, 32, 1, 1, 1, 2, 5, 15, 47, 89, 64, 1, 1, 1, 2, 5, 15, 52, 150, 233, 128, 1, 1, 1, 2, 5, 15, 52, 188, 481, 610, 256, 1, 1, 1, 2, 5, 15, 52, 203, 696, 1545, 1597, 512, 1

Offset: 0

Alois P. Heinz, May 21 2017

The sequence of column k satisfies a linear recurrence with constant coefficients of order 2^(k-1) for k>0.

			A(4,0) = 1: 1|2|3|4.
A(4,1) = 8: 1234, 123|4, 12|34, 12|3|4, 1|234, 1|23|4, 1|2|34, 1|2|3|4.
A(4,2) = 13: 1234, 123|4, 124|3, 12|34, 12|3|4, 134|2, 13|24, 13|2|4, 1|234, 1|23|4, 1|24|3, 1|2|34, 1|2|3|4.
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,  8,  13,  15,  15,  15,  15,  15, ...
  1, 16,  34,  47,  52,  52,  52,  52, ...
  1, 32,  89, 150, 188, 203, 203, 203, ...
  1, 64, 233, 481, 696, 825, 877, 877, ...

Alois P. Heinz, Antidiagonals n = 0..45, flattened
Pierpaolo Natalini, Paolo Emilio Ricci, New Bell-Sheffer Polynomial Sets, Axioms 2018, 7(4), 71.
Wikipedia, Partition of a set

Columns k=0-10 give: A000012, A011782, A001519, A287275, A287276, A287277, A287278, A287279, A287280, A287281, A287282.
Main diagonal gives A000110.
Cf. A287213, A287216, A287417, A287641.

b:= proc(n, k, l) option remember; `if`(n=0, 1, b(n-1, k, map(x->
      `if`(x-n>=k, [][], x), [l[], n]))+add(b(n-1, k, sort(map(x->
      `if`(x-n>=k, [][], x), subsop(j=n, l)))), j=1..nops(l)))
    end:
A:= (n, k)-> b(n, min(k, n-1), []):
seq(seq(A(n, d-n), n=0..d), d=0..12);

b[0, , ] = 1; b[n_, k_, l_List] := b[n, k, l] = b[n - 1, k, If[# - n >= k, Nothing, #]& /@ Append[l, n]] + Sum[b[n - 1, k, Sort[If[# - n >= k, Nothing, #]& /@ ReplacePart[l, j -> n]]], {j, 1, Length[l]}];
A[n_, k_] := b[n, Min[k, n - 1], {}];
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} A287213(n,j).

A287417 Number A(n,k) of set partitions of [n] such that all absolute differences between least elements of consecutive blocks and between consecutive elements within the blocks are not larger than k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 1, 1, 2, 0, 1, 1, 2, 3, 0, 1, 1, 2, 5, 4, 0, 1, 1, 2, 5, 12, 5, 0, 1, 1, 2, 5, 15, 27, 6, 0, 1, 1, 2, 5, 15, 46, 58, 7, 0, 1, 1, 2, 5, 15, 52, 139, 121, 8, 0, 1, 1, 2, 5, 15, 52, 187, 410, 248, 9, 0, 1, 1, 2, 5, 15, 52, 203, 677, 1189, 503, 10, 0

Offset: 0

Alois P. Heinz, May 24 2017

			A(5,3) = 46 = 52 - 6 = A000110(5) - 6 counts all set partitions of [5] except: 1234|5, 15|234, 15|23|4, 15|24|3, 15|2|34, 15|2|3|4.
Square array A(n,k) begins:
  1, 1,   1,   1,   1,   1,   1,   1, ...
  1, 1,   1,   1,   1,   1,   1,   1, ...
  0, 2,   2,   2,   2,   2,   2,   2, ...
  0, 3,   5,   5,   5,   5,   5,   5, ...
  0, 4,  12,  15,  15,  15,  15,  15, ...
  0, 5,  27,  46,  52,  52,  52,  52, ...
  0, 6,  58, 139, 187, 203, 203, 203, ...
  0, 7, 121, 410, 677, 824, 877, 877, ...

Alois P. Heinz, Antidiagonals n = 0..40, flattened
Wikipedia, Partition of a set

Columns k=1-10 give: A028310, A000325, A287582, A287583, A287584, A287585, A287586, A287587, A287588, A287589.
Main diagonal gives A000110.
Cf. A287214, A287216, A287416, A287641.

b:= proc(n, k, l, t) option remember; `if`(n<1, 1, `if`(t-n>k, 0,
       b(n-1, k, map(x-> `if`(x-n>=k, [][], x), [l[], n]), n)) +add(
       b(n-1, k, sort(map(x-> `if`(x-n>=k, [][], x), subsop(j=n, l))),
       `if`(t-n>k, infinity, t)), j=1..nops(l)))
    end:
A:= (n, k)-> b(n, min(k, n-1), [], n):
seq(seq(A(n, d-n), n=0..d), d=0..14);

b[n_, k_, l_, t_] := b[n, k, l, t] = If[n < 1, 1, If[t - n > k, 0, b[n - 1, k, If[# - n >= k, Nothing, #]& /@  Append[l, n], n]] + Sum[b[n - 1, k, Sort[If[# - n >= k, Nothing, #]& /@ ReplacePart[l, j -> n]], If[t - n > k, Infinity, t]], {j, 1, Length[l]}]];
A[n_, k_] := b[n, Min[k, n - 1], {}, n];
Table[A[n, d - n], {d, 0, 14}, { n, 0, d}] // Flatten (* Jean-François Alcover, May 24 2018, translated from Maple *)

A(n,k) = Sum_{j=0..k} A287416(n,j).

A287640 Number T(n,k) of set partitions of [n], where k is minimal such that for all j in [n]: j is member of block b implies b = 1 or at least one of j-1, ..., j-k is member of a block >= b-1; triangle T(n,k), n >= 0, 0 <= k <= max(floor(n/2), n-2), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 4, 1, 13, 1, 1, 41, 9, 1, 1, 131, 59, 11, 1, 1, 428, 344, 88, 15, 1, 1, 1429, 1906, 634, 146, 23, 1, 1, 4861, 10345, 4389, 1231, 280, 39, 1, 1, 16795, 55901, 30006, 9835, 2763, 602, 71, 1, 1, 58785, 303661, 205420, 77178, 25014, 6967, 1408, 135, 1

Offset: 0

Alois P. Heinz, May 28 2017

			T(4,0) = 1: 1234.
T(4,1) = 13: 123|4, 124|3, 12|34, 12|3|4, 134|2, 13|24, 14|23, 1|234, 1|23|4, 14|2|3, 1|24|3, 1|2|34, 1|2|3|4.
T(4,2) = 1: 13|2|4.
T(5,2) = 9: 124|3|5, 135|2|4, 13|25|4, 13|2|45, 13|2|4|5, 14|23|5, 14|2|35, 14|2|3|5, 1|24|3|5.
T(6,3) = 11: 1245|3|6, 1346|2|5, 134|26|5, 134|2|56, 134|2|5|6, 145|23|6, 145|2|36, 145|2|3|6, 14|25|3|6, 15|24|3|6, 1|245|3|6.
T(6,4) = 1: 1345|2|6.
T(7,4) = 15: 12456|3|7, 13457|2|6, 1345|27|6, 1345|2|67, 1345|2|6|7, 1456|23|7, 1456|2|37, 1456|2|3|7, 145|26|3|7, 146|25|3|7, 14|256|3|7, 156|24|3|7, 15|246|3|7, 16|245|3|7, 1|2456|3|7.
Triangle T(n,k) begins:
  1;
  1;
  1,    1;
  1,    4;
  1,   13,     1;
  1,   41,     9,    1;
  1,  131,    59,   11,    1;
  1,  428,   344,   88,   15,   1;
  1, 1429,  1906,  634,  146,  23,  1;
  1, 4861, 10345, 4389, 1231, 280, 39, 1;
  ...

Alois P. Heinz, Rows n = 0..20, flattened
Wikipedia, Partition of a set

Columns k=0-1 give: A000012, A001453.
Row sums give A000110.
Cf. A168415, A287213, A287215, A287416, A287641.

b:= proc(n, l) option remember; `if`(n=0 or l=[], 1, add(b(n-1,
      [seq(max(l[i], j), i=2..nops(l)), j]), j=1..l[1]+1))
    end:
T:= (n, k)-> `if`(k=0, 1, b(n, [0$k])-b(n, [0$k-1])):
seq(seq(T(n, k), k=0..max(n/2, n-2)), n=0..12);

b[n_, l_] := b[n, l] = If[n == 0 || l == {}, 1, Sum[b[n-1, Append[Table[ Max[l[[i]], j], {i, 2, Length[l]}], j]], {j, 1, l[[1]] + 1}]];
T[n_, k_] := If[k == 0, 1, b[n, Table[0, k]] - b[n, Table[0, k - 1]]];
Table[T[n, k], {n, 0, 12}, { k, 0, Max[n/2, n - 2]}] // Flatten (* Jean-François Alcover, May 22 2018, translated from Maple *)

T(n,k) = A287641(n,k) - A287641(n,k-1) for k>0, T(n,0) = 1.

T(n+4,n+1) = A168415(n) for n>0.

A275605 Number of set partitions of [n] such that j is member of block b only if b = 1 or at least one of j-1, j-2 is member of a block >= b-1.

Original entry on oeis.org

1, 1, 2, 5, 15, 51, 191, 773, 3336, 15207, 72697, 362447, 1876392, 10051083, 55544661, 315899245, 1845139684, 11048651523, 67719859612, 424287619507, 2714074517843, 17706680249505, 117704101959444, 796546613501759, 5483490237025393, 38372546811580251

Offset: 0

Benedict W. J. Irwin, Nov 14 2016

nonn

Original name was: The 'AND' Motzkin Numbers.
This sequence consists of the place values from counting in a pattern where the digit is carried if the current place exceeds both the next place plus one and the place after that plus one. (Note that the place "after" a digit is equally described as the digit preceding it, since we write high-order digits first.)
If the "and" logical comparison is changed to "or", then that modified definition produces the Motzkin numbers A001006.
If the definition looks only at the next term, this generates the Catalan numbers A000108.
This is the case k = 2 of a class of sequences, counting sequences where the k-th term is not more than one more than the maximum of the previous k values. The case k = 1 is the Catalan numbers. The limit as k goes to infinity is the Bell numbers A000110. A similar series limiting terms to no more than one more than the minimum of the previous k values has again the Catalan numbers for k = 1, the Motzkin numbers for k = 2, and continues from there. In this case the limit is the all-ones sequence. - Franklin T. Adams-Watters, Mar 14 2017
To get all the sequences of numerals of length n, take all the numerals of length strictly less than n, and pad them on the left with zeros to length n. - Franklin T. Adams-Watters, May 26 2017

			The sequence of numerals starts 0, 1, 10, 11, 12, 100, 101, 102, 110, 111, 112, 120, 121, 122, 123.
To get the numeral following 12, we first increment the final digit: 13. But the digits before the 3 are 0 (implied) and 1, and 3 is greater than either of those by more than 1. So we set the last digit to 0, and increment the previous one: 20. Again, 2 is too large for the two implicit zeros in front of it, so we set it to 0 and increment the preceding digit, an implicit zero; so we get 100, which presents no problems.
The length 3 numerals come from the numerals less than 100: 0, 1, 10, 11, 12. Inserting leading zeros to length 3 gives 000, 001, 010, 011, 012.
The values of 1, 10, 100, 1000, etc. make up the sequence.
a(5) = 51 = 52 - 1 = A000110(5) - 1 counts all set partitions of [5] except: 134|2|5. - _Alois P. Heinz_, May 27 2017

Alois P. Heinz, Table of n, a(n) for n = 0..400
Benedict Irwin, Integer Sequences by Counting Rules.
Zhicong Lin, Restricted inversion sequences and enhanced 3-noncrossing partitions, arXiv:1706.07213 [math.CO], 2017.

Cf. A000108, A000110, A001006.

Column k=2 of A287641.

b:= proc(n, i, j) option remember; `if`(n=0, 1,
      add(b(n-1, max(j, k), k), k=1..i+1))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..30);  # Alois P. Heinz, May 26 2017

SIZ = 30;MAX = 100000;
M = Table[0, {n, 1, SIZ + 2}];
For[i = 0, i <= MAX, i++,sum = 0;For[j = 1, j <= SIZ, j++,sum += M[[j]];]
  If[sum == 1, Print[i]]M[[1]]++;
For[j = 1, j <= SIZ, j++,If[M[[j]] > M[[j + 1]] + 1 && M[[j]] > M[[j + 2]] + 1, M[[j]] = 0; M[[j + 1]]++]]] (* Benedict W. J. Irwin, Nov 14 2016 *)
b[n_, i_, j_] := b[n, i, j] = If[n==0, 1, Sum[b[n-1, Max[j, k], k], {k, 1, i+1} ] ];
a[n_] := b[n, 0, 0];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jun 23 2017, after Alois P. Heinz *)

Edited by Franklin T. Adams-Watters, May 26 2017

More terms and new name from Alois P. Heinz, May 26 2017

A287666 Number of set partitions of [n] such that j is member of block b only if b = 1 or at least one of j-1, ..., j-3 is member of a block >= b-1.

Original entry on oeis.org

1, 1, 2, 5, 15, 52, 202, 861, 3970, 19596, 102703, 567867, 3295439, 19986462, 126231946, 827759525, 5621051650, 39439867696, 285368007479, 2125566382124, 16273261632111, 127881070062521, 1030221084660031, 8498826714433335, 71721238761675612, 618573094313147709

Offset: 0

Alois P. Heinz, May 29 2017

nonn

			a(6) = 202 = 203 - 1 = A000110(6) - 1 counts all set partitions of [6] except: 1345|2|6.

Alois P. Heinz, Table of n, a(n) for n = 0..150
Wikipedia, Partition of a set

Column k=3 of A287641.

Cf. A000110.

b:= proc(n, l) option remember; `if`(n=0, 1, add(b(n-1,
      [seq(max(l[i], j), i=2..nops(l)), j]), j=1..l[1]+1))
    end:
a:= n-> b(n, [0$3]):
seq(a(n), n=0..26);

b[n_, l_] := b[n, l] = If[n == 0, 1, Sum[b[n - 1, Append[Table[Max[l[[i]], j], {i, 2, Length[l]}], j]], {j, 1, l[[1]] + 1}]];
a[n_] := b[n, {0, 0, 0}];
Table[a[n], {n, 0, 26}] (* Jean-François Alcover, May 27 2018, from Maple *)

a(n) = A287641(n,3).

a(n) = A000110(n) for n <= 5.

A287667 Number of set partitions of [n] such that j is member of block b only if b = 1 or at least one of j-1, ..., j-4 is member of a block >= b-1.

Original entry on oeis.org

1, 1, 2, 5, 15, 52, 203, 876, 4116, 20827, 112538, 645045, 3900512, 24769152, 164546915, 1139818861, 8209631792, 61331709492, 474221335902, 3787741281763, 31199052157724, 264605708064825, 2307562757319104, 20666169125398768, 189855243829576499

Offset: 0

Alois P. Heinz, May 29 2017

nonn

			a(7) = 876 = 877 - 1 = A000110(7) - 1 counts all set partitions of [7] except: 13456|2|7.

Alois P. Heinz, Table of n, a(n) for n = 0..90
Wikipedia, Partition of a set

Column k=4 of A287641.

Cf. A000110.

b:= proc(n, l) option remember; `if`(n=0, 1, add(b(n-1,
      [seq(max(l[i], j), i=2..nops(l)), j]), j=1..l[1]+1))
    end:
a:= n-> b(n, [0$4]):
seq(a(n), n=0..26);

b[n_, l_] := b[n, l] = If[n == 0, 1, Sum[b[n - 1, Append[Table[Max[l[[i]], j], {i, 2, Length[l]}], j]], {j, 1, l[[1]] + 1}]];
a[n_] := b[n, Table[0, 4]];
Table[a[n], {n, 0, 26}] (* Jean-François Alcover, May 27 2018, from Maple *)

a(n) = A287641(n,4).

a(n) = A000110(n) for n <= 6.

A287668 Number of set partitions of [n] such that j is member of block b only if b = 1 or at least one of j-1, ..., j-5 is member of a block >= b-1.

Original entry on oeis.org

1, 1, 2, 5, 15, 52, 203, 877, 4139, 21107, 115301, 670059, 4119316, 26665103, 181031235, 1284643851, 9500643629, 73037739470, 582346938182, 4805997066022, 40980051074202, 360452146946076, 3265691382361850, 30435437254066599, 291431082211368120

Offset: 0

Alois P. Heinz, May 29 2017

nonn

			a(8) = 4139 = 4140 - 1 = A000110(8) - 1 counts all set partitions of [8] except: 134567|2|8.

Alois P. Heinz, Table of n, a(n) for n = 0..55

Column k=5 of A287641.

Cf. A000110.

b:= proc(n, l) option remember; `if`(n=0, 1, add(b(n-1,
      [seq(max(l[i], j), i=2..nops(l)), j]), j=1..l[1]+1))
    end:
a:= n-> b(n, [0$5]):
seq(a(n), n=0..24);

b[n_, l_] := b[n, l] = If[n == 0, 1, Sum[b[n - 1, Append[Table[Max[l[[i]], j], {i, 2, Length[l]}], j]], {j, 1, l[[1]] + 1}]];
a[n_] := b[n, Table[0, {5}]];
a /@ Range[0, 24] (* Jean-François Alcover, Dec 11 2020, after Alois P. Heinz *)

a(n) = A287641(n,5).

a(n) = A000110(n) for n <= 7.

A287669 Number of set partitions of [n] such that j is member of block b only if b = 1 or at least one of j-1, ..., j-6 is member of a block >= b-1.

Original entry on oeis.org

1, 1, 2, 5, 15, 52, 203, 877, 4140, 21146, 115903, 677026, 4190648, 27356008, 187573260, 1346289439, 10084570537, 78630320221, 636692795555, 5342949225111, 46381106554291, 415803352327861, 3843867571153341, 36592205230965683, 358266592635074429

Offset: 0

Alois P. Heinz, May 29 2017

nonn

			a(9) = 21146 = 21147 - 1 = A000110(9) - 1 counts all set partitions of [9] except: 1345678|2|9.

Alois P. Heinz, Table of n, a(n) for n = 0..45
Wikipedia, Partition of a set

Column k=6 of A287641.

Cf. A000110.

b:= proc(n, l) option remember; `if`(n=0, 1, add(b(n-1,
      [seq(max(l[i], j), i=2..nops(l)), j]), j=1..l[1]+1))
    end:
a:= n-> b(n, [0$6]):
seq(a(n), n=0..20);

b[n_, l_] := b[n, l] = If[n == 0, 1, Sum[b[n - 1, Append[Table[Max[l[[i]], j], {i, 2, Length[l]}], j]], {j, 1, l[[1]] + 1}]];
a[n_] := b[n, Table[0, 6]];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, May 27 2018, from Maple *)

a(n) = A287641(n,6).

a(n) = A000110(n) for n <= 8.

A287670 Number of set partitions of [n] such that j is member of block b only if b = 1 or at least one of j-1, ..., j-7 is member of a block >= b-1.

Original entry on oeis.org

1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115974, 678434, 4209827, 27578206, 189954361, 1370870811, 10334533723, 81166980407, 662588540048, 5610196619724, 49177794178940, 445536788068643, 4165402700226511, 40131393651398259, 397935154986242021

Offset: 0

Alois P. Heinz, May 29 2017

nonn

			a(10) = 115974 = 115975 - 1 = A000110(10) - 1 counts all set partitions of [10] except: 13456789|2|(10).

Alois P. Heinz, Table of n, a(n) for n = 0..37
Wikipedia, Partition of a set

Column k=7 of A287641.

Cf. A000110.

b:= proc(n, l) option remember; `if`(n=0, 1, add(b(n-1,
      [seq(max(l[i], j), i=2..nops(l)), j]), j=1..l[1]+1))
    end:
a:= n-> b(n, [0$7]):
seq(a(n), n=0..20);

b[n_, l_] := b[n, l] = If[n == 0, 1, Sum[b[n - 1, Append[Table[Max[l[[i]], j], {i, 2, Length[l]}], j]], {j, 1, l[[1]] + 1}]];
a[n_] := b[n, Table[0, 7]];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, May 27 2018, from Maple *)

a(n) = A287641(n,7).

a(n) = A000110(n) for n <= 9.

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.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A287214 Number A(n,k) of set partitions of [n] such that for each block all absolute differences between consecutive elements are <= k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A287417 Number A(n,k) of set partitions of [n] such that all absolute differences between least elements of consecutive blocks and between consecutive elements within the blocks are not larger than k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A287640 Number T(n,k) of set partitions of [n], where k is minimal such that for all j in [n]: j is member of block b implies b = 1 or at least one of j-1, ..., j-k is member of a block >= b-1; triangle T(n,k), n >= 0, 0 <= k <= max(floor(n/2), n-2), read by rows.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A275605 Number of set partitions of [n] such that j is member of block b only if b = 1 or at least one of j-1, j-2 is member of a block >= b-1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

A287666 Number of set partitions of [n] such that j is member of block b only if b = 1 or at least one of j-1, ..., j-3 is member of a block >= b-1.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A287667 Number of set partitions of [n] such that j is member of block b only if b = 1 or at least one of j-1, ..., j-4 is member of a block >= b-1.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula