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 A367969 Number of partitions of [n] whose block maxima sum to k, where k is chosen so as to maximize this number. 4
 1, 1, 1, 2, 5, 11, 37, 129, 431, 1921, 9544, 43844, 223512, 1407320, 8519457, 52422985, 373424140, 2685768084, 20354852852, 160370778238, 1318493838635, 11239312718146, 98700416575613, 916309760098349, 8735277842452542, 84921152781222758, 860903677319960583 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 LINKS Alois P. Heinz, Table of n, a(n) for n = 0..200 Wikipedia, Partition of a set EXAMPLE a(5) = 11 = A367955(5,12) is the largest value in row 5 of A367955 and counts the partitions of [5] having block maxima sum 12: 123|4|5, 124|3|5, 125|3|4, 13|24|5, 13|25|4, 14|23|5, 15|23|4, 14|25|3, 15|24|3, 1|2|34|5, 1|2|35|4. MAPLE b:= proc(n, m) option remember; `if`(n=0, 1, b(n-1, m)*m + expand(x^n*b(n-1, m+1))) end: a:= n-> max(coeffs(b(n, 0))): seq(a(n), n=0..30); # second Maple program: b:= proc(n, i, t) option remember; `if`(i*(i+1)/2 max(seq(b(k, n, 0), k=n..n*(n+1)/2)): seq(a(n), n=0..30); MATHEMATICA b[n_, m_] := b[n, m] = If[n == 0, 1, b[n-1, m]*m + Expand[x^n*b[n-1, m+1]]]; a[n_] := Max[CoefficientList[b[n, 0], x]]; Table[a[n], {n, 0, 30}] (* second program: *) b[n_, i_, t_] := b[n, i, t] = If[i*(i + 1)/2 < n, 0, If[n == 0, t^i, If[t == 0, 0, t*b[n, i - 1, t]] + (t + 1)^Max[0, 2*i - n - 1]*b[n - i, Min[n - i, i - 1], t + 1]]]; a[n_] := If[n == 0, 1, Max[Table[b[k, n, 0], { k, n, n*(n + 1)/2}]]]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Dec 13 2023, after Alois P. Heinz *) CROSSREFS Row maxima of A367955. Cf. A365441, A365903. Sequence in context: A074497 A131581 A195985 * A056301 A001344 A056302 Adjacent sequences: A367966 A367967 A367968 * A367970 A367971 A367972 KEYWORD nonn AUTHOR Alois P. Heinz, Dec 06 2023 STATUS approved

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Last modified September 13 05:16 EDT 2024. Contains 375859 sequences. (Running on oeis4.)