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A276719 Number A(n,k) of set partitions of [n] such that for each block b the smallest integer interval containing b has at most k elements; square array A(n,k), n>=0, k>=0, read by antidiagonals. 12
1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 2, 3, 1, 0, 1, 1, 2, 5, 5, 1, 0, 1, 1, 2, 5, 10, 8, 1, 0, 1, 1, 2, 5, 15, 20, 13, 1, 0, 1, 1, 2, 5, 15, 37, 42, 21, 1, 0, 1, 1, 2, 5, 15, 52, 87, 87, 34, 1, 0, 1, 1, 2, 5, 15, 52, 151, 208, 179, 55, 1, 0, 1, 1, 2, 5, 15, 52, 203, 409, 515, 370, 89, 1, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,13

COMMENTS

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

LINKS

Alois P. Heinz, Antidiagonals n = 0..40, flattened

Wikipedia, Partition of a set

EXAMPLE

A(3,2) = 3: 12|3, 1|23, 1|2|3.

A(4,3) = 10: 123|4, 12|34, 12|3|4, 13|24, 13|2|4, 1|234, 1|23|4, 1|24|3, 1|2|34, 1|2|3|4.

A(5,4) = 37: 1234|5, 123|45, 123|4|5, 124|35, 124|3|5, 12|345, 12|34|5, 12|35|4, 12|3|45, 12|3|4|5, 134|25, 134|2|5, 13|245, 13|24|5, 13|25|4, 13|2|45, 13|2|4|5, 14|235, 14|23|5, 1|2345, 1|234|5, 1|235|4, 1|23|45, 1|23|4|5, 14|25|3, 14|2|35, 14|2|3|5, 1|245|3, 1|24|35, 1|24|3|5, 1|25|34, 1|2|345, 1|2|34|5, 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, ...

0, 1,  1,   1,   1,    1,    1,    1,    1, ...

0, 1,  2,   2,   2,    2,    2,    2,    2, ...

0, 1,  3,   5,   5,    5,    5,    5,    5, ...

0, 1,  5,  10,  15,   15,   15,   15,   15, ...

0, 1,  8,  20,  37,   52,   52,   52,   52, ...

0, 1, 13,  42,  87,  151,  203,  203,  203, ...

0, 1, 21,  87, 208,  409,  674,  877,  877, ...

0, 1, 34, 179, 515, 1100, 2066, 3263, 4140, ...

MAPLE

b:= proc(n, m, l) option remember; `if`(n=0, 1,

      add(b(n-1, max(m, j), [subsop(1=NULL, l)[],

      `if`(j<=m, 0, j)]), j={l[], m+1} minus {0}))

    end:

A:= (n, k)-> `if`(n=0, 1, `if`(k<2, k, b(n, 0, [0$(k-1)]))):

seq(seq(A(n, d-n), n=0..d), d=0..14);

MATHEMATICA

b[n_, m_, l_List] := b[n, m, l] = If[n == 0, 1, Sum[b[n - 1, Max[m, j], Append[ReplacePart[l, 1 -> Nothing], If[j <= m, 0, j]]], {j, Append[l, m + 1] ~Complement~ {0}}]]; A[n_, k_] := If[n == 0, 1, If[k < 2, k, b[n, 0, Array[0&, k-1]]]]; Table[A[n, d-n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-Fran├žois Alcover, Jan 06 2017, after Alois P. Heinz *)

CROSSREFS

Columns k=0..10 give: A000007, A000012, A000045(n+1), A129847, A276720, A276721, A276722, A276723, A276724, A276725, A276726.

Main diagonal gives A000110.

A(n+1,n) gives A005493(n-1) for n>0.

Cf. A276727, A276837.

Sequence in context: A247506 A182172 A143841 * A276837 A269941 A035440

Adjacent sequences:  A276716 A276717 A276718 * A276720 A276721 A276722

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Sep 16 2016

STATUS

approved

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Last modified February 24 03:21 EST 2018. Contains 299595 sequences. (Running on oeis4.)