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);
-
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 *)
-
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
A336512
Total sum of the left-to-right minima in all compositions of n.
Original entry on oeis.org
0, 1, 3, 8, 17, 38, 78, 162, 330, 672, 1355, 2736, 5503, 11058, 22191, 44507, 89198, 178697, 357852, 716440, 1434041, 2869935, 5742801, 11490298, 22988084, 45988166, 91995547, 184021931, 368093352, 736266262, 1472660452, 2945526806, 5891385159, 11783304479
Offset: 0
a(4) = 1 + 1 + 1 + 2 + 1 + 2 + 1 + 3 + 1 + 4 = 17: (1)111, (1)12, (1)21, (2)(1)1, (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 b(n, n+1)[2]:
seq(a(n), n=0..50);
-
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, Min[m, j]]], {j, 1, n}]];
a[n_] := b[n, n + 1][[2]];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Mar 07 2022, after Alois P. Heinz *)
A336516
Sum of parts, counted without multiplicity, in all compositions of n.
Original entry on oeis.org
0, 1, 3, 10, 24, 59, 136, 309, 682, 1493, 3223, 6904, 14675, 31013, 65202, 136512, 284748, 592082, 1227709, 2539516, 5241640, 10798133, 22206568, 45597489, 93495667, 191464970, 391636718, 800233551, 1633530732, 3331568080, 6789078236, 13824212219, 28129459098
Offset: 0
a(4) = 1 + 1 + 2 + 1 + 2 + 1 + 2 + 2 + 1 + 3 + 3 + 1 + 4 = 24: (1)111, (1)1(2), (1)(2)1, (2)(1)1, (2)2, (1)(3), (3)(1), (4).
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b:= proc(n, i, p) option remember; `if`(n=0, [p!, 0],
`if`(i<1, 0, add((p-> [0, `if`(j=0, 0, p[1]*i)]+p)(
b(n-i*j, i-1, p+j)/j!), j=0..n/i)))
end:
a:= n-> b(n$2, 0)[2]:
seq(a(n), n=0..38);
-
b[n_, i_, p_] := b[n, i, p] = If[n == 0, {p!, 0},
If[i < 1, {0, 0}, Sum[Function[{0, If[j == 0, 0, #[[1]]*i]} + #][
b[n - i*j, i - 1, p + j]/j!], {j, 0, n/i}]]];
a[n_] := b[n, n, 0][[2]];
Table[a[n], {n, 0, 38}] (* Jean-François Alcover, Mar 11 2022, 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 1-4 of 4 results.