A174455 -id:A174455 - cp's OEIS frontend

A182713 Number of 3's in the last section of the set of partitions of n.

0, 0, 1, 0, 1, 2, 2, 3, 6, 6, 10, 14, 18, 24, 35, 42, 58, 76, 97, 124, 164, 202, 261, 329, 412, 514, 649, 795, 992, 1223, 1503, 1839, 2262, 2741, 3346, 4056, 4908, 5919, 7150, 8568, 10297, 12320, 14721, 17542, 20911, 24808, 29456, 34870, 41232, 48652, 57389

Offset: 1

Omar E. Pol, Nov 28 2010

nonn

Also number of 3's in all partitions of n that do not contain 1 as a part.
Also 0 together with the first differences of A024787. - Omar E. Pol, Nov 13 2011
a(n) is the number of partitions of n having fewer 1s than 2s; e.g., a(7) counts these 3 partitions: [5, 2], [3, 2, 2], [2, 2, 2, 1]. - Clark Kimberling, Mar 31 2014
The last section of the set of partitions of n is also the n-th section of the set of partitions of any integer >= n. - Omar E. Pol, Apr 07 2014

			a(7) = 2 counts the 3's in 7 = 4+3 = 3+2+2. The 3's in 7 = 3+3+1 = 3+2+1+1 = 3+1+1+1+1 do not count.
From _Omar E. Pol_, Oct 27 2012: (Start)
--------------------------------------
Last section                   Number
of the set of                    of
partitions of 7                 3's
--------------------------------------
7 .............................. 0
4 + 3 .......................... 1
5 + 2 .......................... 0
3 + 2 + 2 ...................... 1
.   1 .......................... 0
.       1 ...................... 0
.       1 ...................... 0
.           1 .................. 0
.       1 ...................... 0
.           1 .................. 0
.           1 .................. 0
.               1 .............. 0
.               1 .............. 0
.                   1 .......... 0
.                       1 ...... 0
------------------------------------
.       5 - 3 =                  2
.
In the last section of the set of partitions of 7 the difference between the sum of the third column and the sum of the fourth column is 5 - 3 = 2 equaling the number of 3's, so a(7) = 2 (see also A024787).
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Column 3 of A194812.

Cf. A135010, A138121, A174455, A182703, A182712, A182714, A240056.

b:= proc(n, i) option remember; local g, h;
      if n=0 then [1, 0]
    elif i<2 then [0, 0]
    else g:= b(n, i-1); h:= `if`(i>n, [0, 0], b(n-i, i));
         [g[1]+h[1], g[2]+h[2]+`if`(i=3, h[1], 0)]
      fi
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=1..70);  # Alois P. Heinz, Mar 18 2012

z = 60; f[n_] := f[n] = IntegerPartitions[n]; t1 = Table[Count[f[n], p_ /; Count[p, 1] < Count[p, 2]], {n, 0, z}] (* Clark Kimberling, Mar 31 2014 *)
b[n_, i_] := b[n, i] = Module[{g, h}, If[n == 0, {1, 0}, If[i<2, {0, 0}, g = b[n, i-1]; h = If[i>n, {0, 0}, b[n-i, i]]; Join[g[[1]] + h[[1]], g[[2]] + h[[2]] + If[i == 3, h[[1]], 0]]]]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 1, 70}] (* Jean-François Alcover, Nov 30 2015, after Alois P. Heinz *)
Table[Count[Flatten@Cases[IntegerPartitions[n], x_ /; Last[x] != 1], 3], {n, 51}] (* Robert Price, May 15 2020 *)

A182713 = lambda n: sum(list(p).count(3) for p in Partitions(n) if 1 not in p) # D. S. McNeil, Nov 29 2010

It appears that A000041(n) = a(n+1) + a(n+2) + a(n+3), n >= 0. - Omar E. Pol, Feb 04 2012

a(n) ~ A000041(n)/3 ~ exp(Pi*sqrt(2*n/3)) / (12*sqrt(3)*n). - Vaclav Kotesovec, Jan 03 2019

A220504 Triangle read by rows: T(n,k) is the total number of appearances of k as the smallest part in all partitions of n.

Original entry on oeis.org

1, 2, 1, 4, 0, 1, 7, 2, 0, 1, 12, 1, 0, 0, 1, 19, 4, 2, 0, 0, 1, 30, 3, 1, 0, 0, 0, 1, 45, 8, 1, 2, 0, 0, 0, 1, 67, 7, 4, 1, 0, 0, 0, 0, 1, 97, 15, 3, 1, 2, 0, 0, 0, 0, 1, 139, 15, 4, 1, 1, 0, 0, 0, 0, 0, 1, 195, 27, 8, 4, 1, 2, 0, 0, 0, 0, 0, 1, 272, 29, 8, 3, 1, 1, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Omar E. Pol, Jan 19 2013

In other words, T(n,k) is the total number of appearances of k in all partitions of n whose smallest part is k.
The sum of row n equals spt(n), the smallest part partition function (see A092269).
T(n,k) is also the sum of row k in the slice n of tetrahedron A209314.

			Triangle begins:
    1;
    2,  1;
    4,  0, 1;
    7,  2, 0, 1;
   12,  1, 0, 0, 1;
   19,  4, 2, 0, 0, 1;
   30,  3, 1, 0, 0, 0, 1;
   45,  8, 1, 2, 0, 0, 0, 1;
   67,  7, 4, 1, 0, 0, 0, 0, 1;
   97, 15, 3, 1, 2, 0, 0, 0, 0, 1;
  139, 15, 4, 1, 1, 0, 0, 0, 0, 0, 1;
  195, 27, 8, 4, 1, 2, 0, 0, 0, 0, 0, 1;
  272, 29, 8, 3, 1, 1, 0, 0, 0, 0, 0, 0, 1;
  ...
The partitions of 6 with the smallest part in brackets are
..........................
.                      [6]
..........................
.                  [3]+[3]
..........................
.                   4 +[2]
.              [2]+[2]+[2]
..........................
.                   5 +[1]
.               3 + 2 +[1]
.               4 +[1]+[1]
.           2 + 2 +[1]+[1]
.           3 +[1]+[1]+[1]
.       2 +[1]+[1]+[1]+[1]
.  [1]+[1]+[1]+[1]+[1]+[1]
..........................
There are 19 smallest parts of size 1. Also there are four smallest parts of size 2. Also there are two smallest parts of size 3. There are no smallest part of size 4 or 5. Finally there is only one smallest part of size 6. So row 6 gives 19, 4, 2, 0, 0, 1. The sum of row 6 is 19+4+2+0+0+1 = A092269(6) = 26.

Alois P. Heinz, Rows n = 1..141, flattened

Columns 1-3: A000070, A087787, A174455.
Row sums give A092269.
Cf. A026794, A182703, A209314.

b:= proc(n, i) option remember; local j, r; if n=0 or i<1 then 0
      else `if`(irem(n, i, 'r')=0, [0$(i-1), r], []); for j from 0
      to n/i do zip((x, y)->x+y, %, [b(n-i*j, i-1)], 0) od; %[] fi
    end:
T:= n-> b(n, n):
seq(T(n), n=1..20);  # Alois P. Heinz, Jan 20 2013

b[n_, i_] := b[n, i] = Module[{j, q, r, pc}, If [n == 0 || i<1, 0, {q, r} = QuotientRemainder[n, i]; pc = If[r == 0, Append[Array[0&, i-1], q], {}]; For[j = 0, j <= n/i, j++, pc = Plus @@ PadRight[{pc, b[n-i*j, i-1]}]]; pc]]; T[n_] := b[n, n]; Table[T[n], {n, 1, 20}] // Flatten (* Jean-François Alcover, Jan 30 2014, after Alois P. Heinz *)

A240056 Number of partitions of n such that m(1) > m(2), where m = multiplicity.

Original entry on oeis.org

0, 1, 1, 1, 3, 4, 5, 9, 12, 16, 24, 32, 42, 59, 77, 100, 134, 173, 221, 288, 366, 463, 590, 741, 926, 1163, 1444, 1787, 2215, 2726, 3342, 4101, 5003, 6087, 7402, 8964, 10827, 13069, 15718, 18865, 22617, 27041, 32263, 38453, 45719, 54264, 64326, 76102, 89884

Offset: 0

Clark Kimberling, Mar 31 2014

			a(7) counts these 9 partitions: 61, 511, 4111, 331, 3211, 31111, 22111, 211111, 1111111.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)

Cf. A182713, A174455, A000041.

z = 60; f[n_] := f[n] = IntegerPartitions[n]; t1 = Table[Count[f[n], p_ /; Count[p, 1] < Count[p, 2]], {n, 0, z}]  (* A182713 *)
t2 = Table[Count[f[n], p_ /; Count[p, 1] <= Count[p, 2]], {n, 0, z}] (* A182713(n+2) *)
t3 = Table[Count[f[n], p_ /; Count[p, 1] == Count[p, 2]], {n, 0, z}] (* A174455 *)
t4 = Table[Count[f[n], p_ /; Count[p, 1] > Count[p, 2]], {n, 0, z}]  (* A240056 *)
t5 = Table[Count[f[n], p_ /; Count[p, 1] >= Count[p, 2]], {n, 0, z}] (* A240056(n+1) *)

a(n) = A000041(n) - A182713(n+2) = a(n+1) - A174455(n) for n >= 0.

a(n) ~ exp(sqrt(2*n/3)*Pi) / (2 * 3^(3/2) * n). - Vaclav Kotesovec, Jan 15 2022

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A182713 Number of 3's in the last section of the set of partitions of n.

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A220504 Triangle read by rows: T(n,k) is the total number of appearances of k as the smallest part in all partitions of n.

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A240056 Number of partitions of n such that m(1) > m(2), where m = multiplicity.

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