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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. 5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,6

LINKS

Alois P. Heinz, Rows n = 0..20, flattened

Wikipedia, Partition of a set

FORMULA

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.

EXAMPLE

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;

MAPLE

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);

MATHEMATICA

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-Fran├žois Alcover, May 22 2018, translated from Maple *)

CROSSREFS

Columns k=0-1 give: A000012, A001453.

Row sums give A000110.

Cf. A168415, A287213, A287215, A287416, A287641.

Sequence in context: A157404 A135704 A002564 * A322078 A019428 A303547

Adjacent sequences:  A287637 A287638 A287639 * A287641 A287642 A287643

KEYWORD

nonn,tabf

AUTHOR

Alois P. Heinz, May 28 2017

STATUS

approved

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Last modified November 14 12:33 EST 2019. Contains 329114 sequences. (Running on oeis4.)