A062023 -id:A062023 - cp's OEIS frontend

A229078 Number of ascending runs in {1,...,n}^n.

0, 1, 7, 63, 736, 10625, 182736, 3647119, 82837504, 2109289329, 59500000000, 1841557146671, 62041198952448, 2259914256880657, 88499197217837056, 3707501605224609375, 165444235911082541056, 7834451891982365825441, 392371124973096027488256

Offset: 0

Alois P. Heinz, Sep 12 2013

nonn

			a(1) = 1: [1].
a(2) = 7 = 2+2+1+2: [1,1], [2,1], [1,2], [2,2].

Alois P. Heinz, Table of n, a(n) for n = 0..200
Eric Weisstein's World of Mathematics, Lambert W-Function
Wikipedia, Lambert W function

Main diagonal of A229079.

Cf. A062023 (nondescending runs), A066274.

a:= n-> `if`(n=0, 0, n^(n-1)*(n*(n+2)-1)/2):
seq(a(n), n=0..25);

a(n) = n^(n-1)*(n*(n+2)-1)/2 for n>0, a(0) = 0.
E.g.f.: 1/2*W(-x)*(W(-x)^3+W(-x)^2-W(-x)-2)/(1+W(-x))^3, W(x) Lambert's function (principal branch).
a(n) = A062023(n) + A066274(n) for n>0.

A225753 Triangle of transformations with k monotonic runs.

Original entry on oeis.org

1, 3, 1, 10, 16, 1, 35, 155, 65, 1, 126, 1246, 1506, 246, 1, 462, 9142, 24017, 12117, 917, 1, 1716, 63792, 315918, 349840, 88852, 3424, 1, 6435, 432399, 3707559, 7635987, 4362297, 619677, 12861, 1, 24310, 2881450, 40455910, 140543458, 149803270, 49462810, 4200670, 48610, 1

Offset: 1

Chad Brewbaker, May 14 2013

Analogous to the Eulerian triangle for permutations A173018.
T(n,k) is the number of words of length n over the alphabet {0,1,...,n-1} that have k-1 descents, see example. [Joerg Arndt, Jun 25 2013]
The expected number of descents is (Sum_{k=1..n} (k-1)*T(n,k)) / (Sum_{k=1..n} T(n,k)) = (n + 1/n -2)/2. - Geoffrey Critzer, Jun 26 2013

			T(1,1) = #{[0]} = 1.
T(2,1) = #{[0,0], [0,1], [1,1]} = 3.
T(2,2) = #{[1,0]} = 1.
T(3,1) = #{[0,0,0], [0,0,1], [0,0,2], [0,1,1], [0,1,2], [0,2,2], [1,1,1], [1,1,2], [1,2,2], [2,2,2]} = 10.
Triangle T(n,k) begins:
     1;
     3,     1;
    10,    16,      1;
    35,   155,     65,      1;
   126,  1246,   1506,    246,     1;
   462,  9142,  24017,  12117,   917,    1;
  1716, 63792, 315918, 349840, 88852, 3424,  1;
  ...

Alois P. Heinz, Rows n = 1..100, flattened

First column is A001700(n-1).
Row sums give: A000312.
Cf. A008292, A062023, A190295, A226998, A333829.

b:= proc(n, l, k) option remember; local j;
      if n=0 then [1] else []; for j to k do zip((x, y)->x+y,
       %, [`if`(j b(n, 0, n)[]:
seq(T(n), n=1..10);  # Alois P. Heinz, Jun 26 2013

Table[Distribution[Map[Length,Map[Split[#,LessEqual[#1,#2]&]&,Tuples[Range[1,n],n]]]],{n,1,7}]//Grid (* Geoffrey Critzer, Jun 25 2013 *)
zip[f_, x_, y_, z_] := With[{m = Max[Length[x], Length[y]]}, f[PadRight[x, m, z], PadRight[y, m, z]]];
b[n_, l_, k_] := b[n, l, k] = Module[{j, pc}, If[n == 0, {1}, pc = {}; For[j = 1, j <= k, j++, pc = zip[Plus, pc, Join[If[jJean-François Alcover, Dec 05 2023, after Alois P. Heinz *)

A226998 The number of descents over all functions f:{1,2,...,n}->{1,2,...,n}.

Original entry on oeis.org

0, 1, 18, 288, 5000, 97200, 2117682, 51380224, 1377495072, 40500000000, 1296871230050, 44952006426624, 1677462128818632, 67068898339975168, 2860906750488281250, 129703669268270284800, 6228632560085359165568, 315864220382241648869376, 16868630748261621128320242

Offset: 1

Geoffrey Critzer, Jun 26 2013

nonn

A descent is an element j in {1,2,...,n-1} such that f(j) > f(j+1).

Cf. A225753.

Mathematica
```
Table[(n + 1/n -2)*n^n/2,{n,1,20}]
```

a(n) = (n + 1/n -2)*n^n/2 = A062023(n) - n^n.

cp's OEIS Frontend

A229078 Number of ascending runs in {1,...,n}^n.

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A225753 Triangle of transformations with k monotonic runs.

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A226998 The number of descents over all functions f:{1,2,...,n}->{1,2,...,n}.

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Mathematica

Formula