A336511
Total sum of the left-to-right maxima in all compositions of n.
Original entry on oeis.org
0, 1, 3, 9, 22, 52, 117, 260, 565, 1217, 2593, 5487, 11538, 24146, 50316, 104490, 216337, 446754, 920506, 1892904, 3885719, 7964162, 16300646, 33321640, 68038796, 138784403, 282824924, 575866839, 1171612786, 2381938742, 4839331484, 9825841526, 19938975797
Offset: 0
a(4) = 1 + 1 + 2 + 1 + 2 + 2 + 2 + 1 + 3 + 3 + 4 = 22: (1)111, (1)1(2), (1)(2)1, (2)11, (2)2, (1)(3), (3)1, (4).
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b:= proc(n, m) option remember; `if`(n=0, [1, 0], add((p-> [0,
`if`(j>m, j*p[1], 0)]+p)(b(n-j, max(m, j))), j=1..n))
end:
a:= n-> b(n, -1)[2]:
seq(a(n), n=0..50);
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b[n_, m_] := b[n, m] = If[n == 0, {1, 0}, Sum[Function[p, {0,
If[j > m, j*p[[1]], 0]} + p][b[n - j, Max[m, j]]], {j, 1, n}]];
a[n_] := b[n, -1][[2]];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Mar 04 2022, after Alois P. Heinz *)
A336718
Total number of left-to-right maxima in all compositions of n into distinct parts.
Original entry on oeis.org
0, 1, 1, 4, 4, 7, 18, 21, 32, 46, 107, 121, 193, 257, 379, 728, 900, 1299, 1806, 2529, 3360, 6182, 7387, 10807, 14385, 20217, 26207, 36450, 58194, 72887, 101130, 135379, 183178, 240796, 323307, 417625, 649959, 797623, 1096645, 1426108, 1931340, 2470541
Offset: 0
a(6) = 18 = 3+2+2+2+1+1+2+1+2+1+1: (1)(2)(3), (1)(3)2, (2)1(3), (2)(3)1, (3)12, (3)21, (2)(4), (4)2, (1)(5), (5)1, (6).
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g:= proc(n) option remember;
`if`(n<2, n, (2*n-1)*g(n-1)-(n-1)^2*g(n-2))
end:
b:= proc(n, i, p) option remember; `if`(i*(i+1)/2 b(n$2, 0):
seq(a(n), n=0..50);
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g[n_] := g[n] = If[n < 2, n, (2 n - 1) g[n - 1] - (n - 1)^2 g[n - 2]];
b[n_, i_, p_] := b[n, i, p] = If[i (i + 1)/2 < n, 0, If[n == 0, g[p], b[n, i - 1, p] + b[n - i, Min[n - i, i - 1], p + 1]]];
a[n_] := b[n, n, 0];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jul 12 2021, after Alois P. Heinz *)
A336770
Total sum of the left-to-right minima in all compositions of n into distinct parts.
Original entry on oeis.org
0, 1, 2, 7, 9, 18, 39, 54, 83, 133, 268, 337, 542, 754, 1148, 2058, 2689, 3909, 5607, 7945, 10965, 19024, 23838, 34840, 47332, 67121, 89006, 125571, 194513, 250634, 349001, 473018, 644107, 860595, 1164018, 1532321, 2327654, 2923772, 4022746, 5290310, 7188111
Offset: 0
a(6) = 39 = 1+1+3+3+4+6+2+6+1+6+6: (1)23, (1)32, (2)(1)3, (2)3(1), (3)(1)2, (3)(2)(1), (2)4, (4)(2), (1)5, (5)(1), (6).
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b:= proc(n, i, k) option remember; `if`(i
(2*i-k+1)*k/2, 0, `if`(n=0, [1, 0], b(n, i-1, k)+
(p-> p+[0, p[1]*i/k])(b(n-i, min(n-i, i-1), k-1))))
end:
a:= n-> add(b(n$2, k)[2]*k!, k=1..floor((sqrt(8*n+1)-1)/2)):
seq(a(n), n=0..40);
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b[n_, i_, k_] := b[n, i, k] = If[i < k || n > (2i - k + 1) k/2, {0, 0}, If[n == 0, {1, 0}, b[n, i - 1, k] + Function[p, p + {0, p[[1]] i/k}][b[n - i, Min[n - i, i - 1], k - 1]]]];
a[n_] := Sum[b[n, n, k][[2]] k!, {k, 1, Floor[(Sqrt[8n + 1] - 1)/2]}];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Jul 12 2021, after Alois P. Heinz *)
Showing 1-3 of 3 results.
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