A336718 -id:A336718

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A336482

Total number of left-to-right maxima in all compositions of n.

+10
5

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000

EXAMPLE

a(4) = 11: (1)111, (1)1(2), (1)(2)1, (2)11, (2)2, (1)(3), (3)1, (4).

MAPLE

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);

MATHEMATICA

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 *)

CROSSREFS

Cf. A000254 (the same for permutations of [n]), A225095, A336484, A336511, A336718.

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Jul 22 2020

STATUS

approved

A336770

Total sum of the left-to-right minima in all compositions of n into distinct parts.

+10
4

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000

EXAMPLE

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).

MAPLE

b:= proc(n, i, k) option remember; `if`(i<k or n>

(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);

MATHEMATICA

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 *)

CROSSREFS

Cf. A000009, A000254, A003056, A032020, A336512, A336718, A336771.

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Aug 04 2020

STATUS

approved

A336771

Total sum of the left-to-right maxima in all compositions of n into distinct parts.

+10
4

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000

EXAMPLE

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).

MAPLE

b:= proc(n, i, k, m) option remember; `if`(i<k or n>

(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);

MATHEMATICA

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 *)

CROSSREFS

Cf. A000009, A000254, A003056, A032020, A336511, A336718, A336770.

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Aug 04 2020

STATUS

approved

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