A287416 -id:A287416 - cp's OEIS frontend

A287215 Number T(n,k) of set partitions of [n] such that the maximal absolute difference between the least elements of consecutive blocks equals k; triangle T(n,k), n>=0, 0<=k<=max(n-1,0), read by rows.

1, 1, 1, 1, 1, 3, 1, 1, 8, 5, 1, 1, 22, 21, 7, 1, 1, 65, 86, 39, 11, 1, 1, 209, 361, 209, 77, 19, 1, 1, 732, 1584, 1123, 493, 171, 35, 1, 1, 2780, 7315, 6153, 3124, 1293, 413, 67, 1, 1, 11377, 35635, 34723, 20019, 9320, 3709, 1059, 131, 1, 1, 49863, 183080, 202852, 130916, 66992, 30396, 11373, 2837, 259, 1

Offset: 0

Alois P. Heinz, May 21 2017

The maximal absolute difference is assumed to be zero if there are fewer than two blocks.

T(n,k) is defined for all n,k >= 0. The triangle contains only the positive terms. T(n,k) = 0 if k>=n and k>0.

			T(4,0) = 1: 1234.
T(4,1) = 8: 134|2, 13|24, 14|23, 1|234, 14|2|3, 1|24|3, 1|2|34, 1|2|3|4.
T(4,2) = 5: 124|3, 12|34, 12|3|4, 13|2|4, 1|23|4.
T(4,3) = 1: 123|4.
Triangle T(n,k) begins:
  1;
  1;
  1,   1;
  1,   3,    1;
  1,   8,    5,    1;
  1,  22,   21,    7,   1;
  1,  65,   86,   39,  11,   1;
  1, 209,  361,  209,  77,  19,  1;
  1, 732, 1584, 1123, 493, 171, 35, 1;

Alois P. Heinz, Rows n = 0..141, flattened
Wikipedia, Partition of a set

Columns k=0-10 give: A000012, A003101(n-1), A322875, A322876, A322877, A322878, A322879, A322880, A322881, A322882, A322883.
Row sums give A000110.
T(2n,n) gives A322884.
Cf. A287213, A287216, A287416, A287640.

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):
T:= (n, k)-> A(n, k)-`if`(k=0, 0, A(n, k-1)):
seq(seq(T(n, k), k=0..max(n-1, 0)), n=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];
T[n_, k_] := A[n, k] - If[k == 0, 0, A[n, k - 1]];
Table[T[n, k], {n, 0, 12}, {k, 0, Max[n - 1, 0]}] // Flatten (* Jean-François Alcover, May 19 2018, after Alois P. Heinz *)

T(n,k) = A287216(n,k) - A287216(n,k-1) for k>0, T(n,0) = 1.

A287213 Number T(n,k) of set partitions of [n] such that the maximal absolute difference between consecutive elements within a block equals k; triangle T(n,k), n>=0, 0<=k<=max(n-1,0), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 7, 5, 2, 1, 15, 18, 13, 5, 1, 31, 57, 61, 38, 15, 1, 63, 169, 248, 215, 129, 52, 1, 127, 482, 935, 1061, 836, 495, 203, 1, 255, 1341, 3368, 4835, 4789, 3573, 2108, 877, 1, 511, 3669, 11777, 20973, 25430, 22986, 16657, 9831, 4140

Offset: 0

Alois P. Heinz, May 21 2017

The maximal absolute difference is assumed to be zero if there is no block with consecutive elements.

T(n,k) is defined for all n,k >= 0. The triangle contains only the positive terms. T(n,k) = 0 if k>=n and k>0.

			T(4,0) = 1: 1|2|3|4.
T(4,1) = 7: 1234, 123|4, 12|34, 12|3|4, 1|234, 1|23|4, 1|2|34.
T(4,2) = 5: 124|3, 134|2, 13|24, 13|2|4, 1|24|3.
T(4,3) = 2: 14|23, 14|2|3.
Triangle T(n,k) begins:
  1;
  1;
  1,   1;
  1,   3,   1;
  1,   7,   5,   2;
  1,  15,  18,  13,    5;
  1,  31,  57,  61,   38,  15;
  1,  63, 169, 248,  215, 129,  52;
  1, 127, 482, 935, 1061, 836, 495, 203;

Alois P. Heinz, Rows n = 0..23, flattened
Wikipedia, Partition of a set

Columns k=0-10 give: A000012, A000225(n-1), A258109, A294052, A294053, A294054, A294055, A294056, A294057, A294058, A294059.
Row sums and T(n+2,n+1) give A000110.
T(2n,n) gives A294024.
Cf. A287214, A287215, A287416, A287640.

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), []):
T:= (n, k)-> A(n, k)-`if`(k=0, 0, A(n, k-1)):
seq(seq(T(n, k), k=0..max(n-1, 0)), n=0..12);

b[0, , ] = 1; b[n_, k_, l_] := 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], {}];
T[n_, k_] :=  A[n, k] - If[k == 0, 0, A[n, k - 1]];
Table[Table[T[n, k], {k, 0, Max[n - 1, 0]}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Apr 30 2018, after Alois P. Heinz *)

T(n,k) = A287214(n,k) - A287214(n,k-1) for k>0, T(n,0) = 1.

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.

cp's OEIS Frontend

A287215 Number T(n,k) of set partitions of [n] such that the maximal absolute difference between the least elements of consecutive blocks equals k; triangle T(n,k), n>=0, 0<=k<=max(n-1,0), read by rows.

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A287213 Number T(n,k) of set partitions of [n] such that the maximal absolute difference between consecutive elements within a block equals k; triangle T(n,k), n>=0, 0<=k<=max(n-1,0), read by rows.

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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.

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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.

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