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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 Pierpaolo Natalini, Paolo Emilio Ricci, New Bell-Sheffer Polynomial Sets, Axioms 2018, 7(4), 71. Wikipedia, Partition of a set FORMULA A(n,k) = Sum_{i=0..k} A276727(n,i). 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-François 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 December 1 22:12 EST 2021. Contains 349435 sequences. (Running on oeis4.)