A336482
Total number of left-to-right maxima in all compositions of n.
Original entry on oeis.org
0, 1, 2, 5, 11, 24, 51, 108, 226, 471, 976, 2015, 4146, 8508, 17418, 35590, 72597, 147868, 300797, 611202, 1240690, 2516268, 5099242, 10326282, 20897848, 42267257, 85442478, 172635651, 348651294, 703836046, 1420315254, 2865122304, 5777735296, 11647641296
Offset: 0
a(4) = 11: (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, c) option remember; `if`(n=0, c, add(
b(n-j, max(m, j), c+`if`(j>m, 1, 0)), j=1..n))
end:
a:= n-> b(n, -1, 0):
seq(a(n), n=0..50);
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b[n_, m_, c_] := b[n, m, c] = If[n == 0, c, Sum[
b[n - j, Max[m, j], c + If[j > m, 1, 0]], {j, 1, n}]];
a[n_] := b[n, -1, 0];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Mar 04 2022, after Alois P. Heinz *)
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T_xy(max_row) = {my(N=max_row+1, x='x+O('x^N), h= prod(i=1,N, 1 + y*x^i *(1-x)/(1-2*x+x^(i+1)))); h}
P_xy(N) = Pol(T_xy(N), {x})
B_x(N) = {my(cx = deriv(P_xy(N), y), y=1); Vecrev(eval(cx))}
B_x(30) \\ John Tyler Rascoe, Mar 22 2025
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 *)
A336771
Total sum of the left-to-right maxima in all compositions of n into distinct parts.
Original entry on oeis.org
0, 1, 2, 8, 11, 22, 53, 75, 123, 193, 418, 538, 894, 1268, 1950, 3567, 4799, 7143, 10355, 14968, 20701, 36398, 46420, 69071, 94972, 136385, 182522, 259104, 402405, 527090, 741569, 1015491, 1397661, 1880541, 2567202, 3392612, 5153156, 6553844, 9088372, 12040797
Offset: 0
a(6) = 53 = 6+4+5+5+3+3+6+4+6+5+6: (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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b:= proc(n, i, k, m) option remember; `if`(i
(2*i-k+1)*k/2, 0, `if`(n=0, [1, 0], b(n, i-1, k, m)+
(p-> p+[0, p[1]*i/(m+1-k)])(b(n-i, min(n-i, i-1), k-1, m))))
end:
a:= n-> add(b(n$2, k$2)[2]*k!, k=1..floor((sqrt(8*n+1)-1)/2)):
seq(a(n), n=0..40);
-
b[n_, i_, k_, m_] := b[n, i, k, m] = If[i < k || n >
(2*i - k + 1)*k/2, {0, 0}, If[n == 0, {1, 0}, b[n, i - 1, k, m] +
Function[p, p+{0, p[[1]]*i/(m+1-k)}][b[n-i, Min[n-i, i-1], k-1, m]]]];
a[n_] := Sum[b[n, n, k, k][[2]]*k!, {k, 1, Floor[(Sqrt[8*n + 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.