A366156 -id:A366156 - cp's OEIS frontend

A366745 Triangular array, read by rows: T(n,k) = number of partitions p of n such that d(p, p') = 2k, where p' = contraconjugate of p, and d is the distance function defined in A366156.

1, 0, 2, 1, 0, 2, 1, 2, 0, 2, 1, 2, 2, 0, 2, 1, 0, 6, 2, 0, 2, 1, 2, 2, 6, 2, 0, 2, 0, 6, 2, 4, 6, 2, 0, 2, 0, 2, 4, 8, 6, 6, 2, 0, 2, 0, 4, 8, 8, 8, 4, 6, 2, 0, 2, 0, 6, 10, 8, 12, 6, 4, 6, 2, 0, 2, 1, 4, 14, 8, 16, 10, 6, 6, 8, 2, 0, 2, 1, 4, 18, 10, 14, 20, 10, 6, 6, 8, 2, 0, 2, 1, 8, 8, 18, 20, 22, 16, 16, 6, 8, 8, 2, 0, 2, 0, 6, 18, 28, 30, 24, 16, 16, 10, 8, 8, 8, 2, 0, 2, 1, 4, 20, 28, 30, 36, 16, 34, 18, 16, 4, 12, 8, 2, 0, 2

Offset: 1

Clark Kimberling, Oct 25 2023

Let RL denote reverse lexicographic. Write each partition p of n in RL order, and then arrange all the partitions of n in RL order, obtaining a list L as in Mathematica's IntegerPartitions[n]. Let L' be the reversal of L. Let p(i) be the i-th partition in L and let p'(i) be the i-th in L'; then p'(i) is introduced here as the contraconjugate of p(i) . (Note that RL order could be called "greedy" order, because the result of applying the greedy algorithm to p is the same as for RL, and likewise for arranging the partitions in RL order. For a discussion of various orderings, see A080577.)

			First 14 rows:
  1
  0    2
  1    0    2
  1    2    0    2
  1    2    2    0    2
  1    0    6    2    0    2
  1    2    2    6    2    0    2
  0    6    2    4    6    2    0    2
  0    2    4    8    6    6    2    0    2
  0    4    8    8    8    4    6    2    0    2
  0    6   10    8   12    6    4    6    2    0    2
  1    4   14    8   16   10    6    6    8    2    0    2
  1    4   18   10   14   20   10    6    6    8    2    0    2
  1    8    8   18   20   22   16   16    6    8    8    2    0    2
The list L for the partitions of 5 is:
  [5], [4,1], [3,2], [3,1,1], [2,2,1], [2,1,1,1], [1,1,1,1,1].
The list L' of contraconjugates p' and distances d(p,p') are given by:
  p = [5]; p' = [1,1,1,1,1]; d(p,p') = 8
  p = [4,1]; p' = [2,1,1,1]; d(p,p') = 4
  p = [3,2]; p' = [2,2,1]; d(p,p') = 2
  p = [3,1,1,1]; p' = [3,1,1,1]; d(p,p') = 0
  p = [2,2,1]; p' = [3,2]; d(p,p') = 2
  p = [2,1,1,1]; p' = [4,1]; d(p,p') = 4
  p = [1,1,1,1,1], p' = [5]; d(p,p') = 8;
so that the numbers of 0s, 2s, 4s, 6s, 8s are 1, 2, 2, 0, 2, as in row 5.

Cf. A000041, A366156, A366746.

c[n_] := PartitionsP[n];
p[n_, k_] := p[n, k] = IntegerPartitions[n][[k]];
r[n_, k_] := r[n, k] = Join[p[n, k], ConstantArray[0, n - Length[p[n, k]]]];
p1[n_, k_] := p1[n, k] = Reverse[IntegerPartitions[n]][[k]];
r1[n_, k_] := r1[n, k] = Join[p1[n, k], ConstantArray[0, n - Length[p1[n, k]]]];
d[u_, v_] := Total[Abs[u - v]];
t[n_] := Flatten[Table[d[r[n, k], r1[n, k]], {k, 1, c[n]}]];
t1 = Table[Count[t[n], m], {n, 1, 16}, {m, 0, 2 n - 1, 2}]
TableForm[t1]   (* array *)
u = Flatten[t1] (*sequence *)

A000097 Number of partitions of n if there are two kinds of 1's and two kinds of 2's.

Original entry on oeis.org

1, 2, 5, 9, 17, 28, 47, 73, 114, 170, 253, 365, 525, 738, 1033, 1422, 1948, 2634, 3545, 4721, 6259, 8227, 10767, 13990, 18105, 23286, 29837, 38028, 48297, 61053, 76926, 96524, 120746, 150487, 187019, 231643, 286152, 352413, 432937, 530383, 648245

Offset: 0

N. J. A. Sloane

Also number of partitions of 2*n with exactly 2 odd parts (offset 1). - Vladeta Jovovic, Jan 12 2005
Also number of transitions from one partition of n+2 to another, where a transition consists of replacing any two parts with their sum. Remove all 1' and 2' from the partition, replacing them with ((number of 2') + 1) and ((number of 1') + (number of 2') + 1); these are the two parts being summed. Number of partitions of n into parts of 2 kinds with at most 2 parts of the second kind, or of n+2 into parts of 2 kinds with exactly 2 parts of the second kind. - Franklin T. Adams-Watters, Mar 20 2006
From Christian Gutschwager (gutschwager(AT)math.uni-hannover.de), Feb 10 2010: (Start)
a(n) is also the number of pairs of partitions of n+2 which differ by only one box (for bijection see the first Gutschwager link).
a(n) is also the number of partitions of n with two parts marked.
a(n) is also the number of partitions of n+1 with two different parts marked. (End)
Convolution of A000041 and A008619. - Vaclav Kotesovec, Aug 18 2015
a(n) = P(/2,n), a particular case of P(/k,n) defined as follows: P(/0,n) = A000041(n) and P(/k,n) = P(/k-1, n) + P(/k-1,n-k) + P(/k-1, n-2k) + ... Also, P(/k,n) = the convolution of A000041 and the partitions of n with exactly k parts, and g.f. P(/k,n) = (g.f. for P(n)) * 1/(1-x)...(1-x^k). - Gregory L. Simay, Mar 22 2018
a(n) is also the sum of binomial(D(p),2) in partitions p of (n+3), where D(p)= number of different sizes of parts in p. - Emily Anible, Apr 03 2018
Also partitions of 2*(n+1) with alternating sum 2. Also partitions of 2*(n+1) with reverse-alternating sum -2 or 2. - Gus Wiseman, Jun 21 2021
Define the distance graph of the partitions of n using the distance function in A366156 as follows: two vertices (partitions) share an edge if and only if the distance between the vertices is 2. Then a(n) is the number of edges in the distance graph of the partitions of n. - Clark Kimberling, Oct 12 2023

			a(3) = 9 because we have 3, 2+1, 2+1', 2'+1, 2'+1', 1+1+1, 1+1+1', 1+1'+1' and 1'+1'+1'.
From _Gus Wiseman_, Jun 22 2021: (Start)
The a(0) = 1 through a(4) = 9 partitions of 2*(n+1) with exactly 2 odd parts:
  (1,1)  (3,1)    (3,3)      (5,3)
         (2,1,1)  (5,1)      (7,1)
                  (3,2,1)    (3,3,2)
                  (4,1,1)    (4,3,1)
                  (2,2,1,1)  (5,2,1)
                             (6,1,1)
                             (3,2,2,1)
                             (4,2,1,1)
                             (2,2,2,1,1)
The a(0) = 1 through a(4) = 9 partitions of 2*(n+1) with alternating sum 2:
  (2)  (3,1)    (4,2)        (5,3)
       (2,1,1)  (2,2,2)      (3,3,2)
                (3,2,1)      (4,3,1)
                (3,1,1,1)    (3,2,2,1)
                (2,1,1,1,1)  (4,2,1,1)
                             (2,2,2,1,1)
                             (3,2,1,1,1)
                             (3,1,1,1,1,1)
                             (2,1,1,1,1,1,1)
(End)

H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958, p. 90.
J. Riordan, Combinatorial Identities, Wiley, 1968, p. 199.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe and Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Christian Gutschwager, The skew diagram poset and components of skew characters, arXiv:1104.0008 [math.CO], 2011.
Christian Gutschwager, Reduced Kronecker products which are multiplicity free or contain only a few components, Eur. J. Combinat. 31 (2010) 1996-2005. doi:10.1016/j.ejc.2010.05.008.
J. P. Robinson, Edges in the poset of partitions of an integer, J. Combin. Theory Ser. A, 48 (1988), 236-238.
N. J. A. Sloane, Transforms

First differences are in A024786.
Cf. A000098, A000710.
Third column of Riordan triangle A008951 and of triangle A103923.
The case of reverse-alternating sum 1 or alternating sum 0 is A000041.
The case of reverse-alternating sum -1 or alternating sum 1 is A000070.
The normal case appears to be A004526 or A065033.
The strict case is A096914.
The case of reverse-alternating sum 2 is A120452.
The case of reverse-alternating sum -2 is A344741.
A001700 counts compositions with alternating sum 2.
A035363 counts partitions into even parts.
A058696 counts partitions of 2n.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A124754 gives alternating sums of standard compositions (reverse: A344618).
A316524 is the alternating sum of the prime indices of n (reverse: A344616).
A344610 counts partitions by sum and positive reverse-alternating sum.
A344611 counts partitions of 2n with reverse-alternating sum >= 0.
Cf. A006330, A027187, A239830, A306145, A343941, A344607, A344608, A344619, A344650, A344651, A344740.
Shift of A093695.

with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; local d,j; if n=0 then 1 else add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n fi end end: a:= etr(n->`if`(n<3,2,1)): seq(a(n), n=0..40); # Alois P. Heinz, Sep 08 2008

CoefficientList[Series[1/((1 - x) (1 - x^2) Product[1 - x^k, {k, 1, 100}]), {x, 0, 100}], x] (* Ben Branman, Mar 07 2012 *)
etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[j]}]*b[n - j], {j, 1, n}]/n]; b]; a = etr[If[# < 3, 2, 1]&]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Apr 09 2014, after Alois P. Heinz *)
(1/((1 - x) (1 - x^2) QPochhammer[x]) + O[x]^50)[[3]] (* Vladimir Reshetnikov, Nov 22 2016 *)
Table[Length@IntegerPartitions[n,All,Join[{1,2},Range[n]]],{n,0,15}] (* Robert Price, Jul 28 2020 and Jun 21 2021 *)
T[n_, 0] := PartitionsP[n];
T[n_, m_] /; (n >= m (m + 1)/2) := T[n, m] = T[n - m, m - 1] + T[n - m, m];
T[, ] = 0;
a[n_] := T[n + 3, 2];
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, May 30 2021 *)
ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];Table[Length[Select[IntegerPartitions[n],ats[#]==2&]],{n,0,30,2}] (* Gus Wiseman, Jun 21 2021 *)

my(x = 'x + O('x^66)); Vec( 1/((1-x)*(1-x^2)*eta(x)) ) \\ Joerg Arndt, Apr 29 2013

Euler transform of 2 2 1 1 1 1 1...
G.f.: 1/( (1-x) * (1-x^2) * Product_{k>=1} (1-x^k) ).
a(n) = Sum_{j=0..floor(n/2)} A000070(n-2*j), n>=0.
a(n) = A014153(n)/2 + A087787(n)/4 + A000070(n)/4. - Vaclav Kotesovec, Nov 05 2016
a(n) ~ sqrt(3) * exp(Pi*sqrt(2*n/3)) / (4*Pi^2) * (1 + 35*Pi/(24*sqrt(6*n))). - Vaclav Kotesovec, Aug 18 2015, extended Nov 05 2016
a(n) = A120452(n) + A344741(n). - Gus Wiseman, Jun 21 2021

More terms from Pab Ter (pabrlos(AT)yahoo.com), May 04 2004
Edited by Emeric Deutsch, Mar 23 2005
More terms from Franklin T. Adams-Watters, Mar 20 2006
Edited by Charles R Greathouse IV, Apr 20 2010

A366429 a(n) = maximum degree of vertices in the distance graph of the partitions of n.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 8, 8, 12, 13, 14, 14, 15, 20, 21, 22, 22, 23, 23, 30, 31, 32, 32, 33, 33, 34, 42, 43, 44, 44, 45, 45, 46, 46, 56, 57, 58, 58, 59, 59, 60, 60, 60, 72, 73, 74, 74, 75, 75, 76, 76, 76, 77, 90, 91, 92, 92, 93, 93, 94, 94, 94, 95, 95, 110, 111

Offset: 1

Clark Kimberling, Oct 12 2023

nonn

The distance graph of the partitions of n is defined by its edges, specifically, two partitions (i.e. vertices) p and q share an edge if d(p,q) = 2, where d is defined in A366156.

			Enumerate the 7 partitions (= vertices) of 5 as follows:
  1: 5
  2: 4,1
  3: 3,2
  4: 3,1,1
  5: 2,2,1
  6: 2,1,1,1
  7: 1,1,1,1,1
Call q a neighbor of p if d(p,q)=2. The set of neighbors for vertex k, for k = 1..7, is given by
  vertex 1: {2}
  vertex 2: {1,3,4}
  vertex 3: {2,4,5}
  vertex 4: {2,3,5,6}
  vertex 5: {3,4,6}
  vertex 6: {4,5,7}
  vertex 7: {6}
The maximal degree is 4, which is the degree of vertex 4, so that a(5) = 4.

Cf. A000041, A000097 (number of edges in distance graphs), A000267, A002024, A002262, A366156, A366461.

c[n_] := PartitionsP[n];
q[n_, k_] := q[n, k] = IntegerPartitions[n][[k]];
r[n_, k_] := r[n, k] = Join[q[n, k], ConstantArray[0, n - Length[q[n, k]]]];
d[u_, v_] := d[u, v] = Total[Abs[u - v]];
s[n_, k_] := s[n, k] = Select[Range[c[n]], d[r[n, k], r[n, #]] == 2 &]
t[n_] := t[n] = Table[s[n, k], {k, 1, c[n]}]
a[n_] := Max[Map[Length, t[n]]]
Table[a[n], {n, 1, 30}]

from math import isqrt, comb
def A366429(n): return isqrt(n-comb((m:=isqrt(k:=n+1<<1))+(k>m*(m+1)),2)<<2|1)+(r:=(isqrt(k<<2)+1>>1)-1)*(r-1)-1 # Chai Wah Wu, Jun 21 2025

a(n) = 2*binomial(A002024(n+1)-1,2) + A000267(A002262(n)) - 1. - Pontus von Brömssen, Oct 24 2023

More terms from Pontus von Brömssen, Oct 24 2023

A368565 a(n) = number of pairs (p,q) of partitions of n such that d(p,q) < o(p,q), where d and o are distance functions; see Comments.

Original entry on oeis.org

0, 0, 0, 0, 0, 30, 106, 316, 652, 1388, 2618, 5170, 9164, 16790, 29046, 50714, 84732, 143588, 234048, 385210, 617050, 990868, 1558310, 2459300, 3806838, 5900184

Offset: 1

Clark Kimberling, Dec 31 2023

The definition of d depends on the greedy ordering of the partitions p(i) of n; that is, p(1) >= p(2) >= ... >= p(k), where k = A000041(n); see A366156. The ordinal distance o is defined by o(p(i),p(j)) = |i-j|.

			The 5 partitions of 4 are (p(1),p(2),p(3),p(4),p(5)) = (4,21,22,211,1111). The following table shows the 25 pairs d(p(i),q(j)) and o(p(i),q(j)):
          |    4   31   22   211  1111
------------------------------------------------
   4 d    |    0    2    4    4    6
     o    |    0    1    2    3    4
  31 d    |    2    0    2    2    4
     o    |    1    0    1    2    3
  22 d    |    4    2    0    2    4
     o    |    2    1    0    1    2
 211 d    |    4    2    2    0    2
     o    |    3    2    1    0    1
1111 d    |    6    4    4    2    0
     o    |    4    3    2    1    0
The table shows 0 pairs (p,q) for which d(p,q) < o(p,q), so a(4) = 0.

Cf. A000041, A001255, A366156, A368564, A368566.

c[n_] := PartitionsP[n];
q[n_, k_] := q[n, k] = IntegerPartitions[n][[k]];
r[n_, k_] := r[n, k] = Join[q[n, k], ConstantArray[0, n - Length[q[n, k]]]];
d[u_, v_] := Total[Abs[u - v]];
p[n_] := Flatten[Table[d[r[n, j], r[n, k]] - Abs[j - k], {j, 1, c[n]}, {k, 1, c[n]}]];
Table[Count[p[n], 0], {n, 1, 16}]  (* A368565  *)
Table[Length[Select[p[n], Sign[#] == -1 &]], {n, 1,16}]  (* A368566 *)
Table[Length[Select[p[n], Sign[#] == 1 &]], {n, 1, 16}]  (* A368567 *)

A368564(n) + a(n) + A368566(n) = A001255(n) for n >= 1.

A368566 a(n) = number of pairs (p,q) of partitions of n such that d(p,q) > o(p,q), where d and o are distance functions; see Comments.

Original entry on oeis.org

0, 2, 6, 18, 34, 48, 62, 108, 166, 242, 334, 512, 706, 984, 1368, 1876, 2492, 3360, 4422, 5848, 7574, 9792, 12596, 16130, 20412, 25850

Offset: 1

Clark Kimberling, Dec 31 2023

			The 5 partitions of 4 are (p(1),p(2),p(3),p(4),p(5)) = (4,21,22,211,1111). The following table shows the 25 pairs d(p(i),q(j)) and o(p(i),q(j)):
          |    4   31   22   211  1111
------------------------------------------------
   4 d    |    0    2    4    4    6
     o    |    0    1    2    3    4
  31 d    |    2    0    2    2    4
     o    |    1    0    1    2    3
  22 d    |    4    2    0    2    4
     o    |    2    1    0    1    2
 211 d    |    4    2    2    0    2
     o    |    3    2    1    0    1
1111 d    |    6    4    4    2    0
     o    |    4    3    2    1    0
The table shows 18 pairs (p,q) for which d(p,q) > o(p,q), so a(4) = 18.

Cf. A000041, A001255, A366156, A368564, A368565.

c[n_] := PartitionsP[n];
q[n_, k_] := q[n, k] = IntegerPartitions[n][[k]];
r[n_, k_] := r[n, k] = Join[q[n, k], ConstantArray[0, n - Length[q[n, k]]]];
d[u_, v_] := Total[Abs[u - v]];
p[n_] := Flatten[Table[d[r[n, j], r[n, k]] - Abs[j - k], {j, 1, c[n]}, {k, 1, c[n]}]];
Table[Count[p[n], 0], {n, 1, 16}]  (* A368565  *)
Table[Length[Select[p[n], Sign[#] == -1 &]], {n, 1,16}]  (* A368566 *)
Table[Length[Select[p[n], Sign[#] == 1 &]], {n, 1, 16}]  (* A368567 *)

A368564(n) + A368565(n) + a(n) = A001255(n) for n >= 1.

A368564 a(n) = number of pairs (p,q) of partitions of n such that d(p,q) = o(p,q), where d and o are distance functions; see Comments.

Original entry on oeis.org

1, 2, 3, 7, 15, 43, 57, 60, 82, 134, 184, 247, 331, 451, 562, 771, 985, 1277, 1630, 2071, 2640, 3344, 4119, 5195, 6514, 8062

Offset: 1

Clark Kimberling, Dec 31 2023

			The 5 partitions of 4 are (p(1),p(2),p(3),p(4),p(5)) = (4,21,22,211,1111). The following table shows the 25 pairs d(p(i),q(j)) and o(p(i),q(j)):
          |    4   31   22   211  1111
------------------------------------------------
   4 d    |    0    2    4    4    6
     o    |    0    1    2    3    4
  31 d    |    2    0    2    2    4
     o    |    1    0    1    2    3
  22 d    |    4    2    0    2    4
     o    |    2    1    0    1    2
 211 d    |    4    2    2    0    2
     o    |    3    2    1    0    1
1111 d    |    6    4    4    2    0
     o    |    4    3    2    1    0
The table shows 7 pairs (p,q) for which d(p,q) = o(p,q), so a(4) = 7.

Cf. A000041, A001255, A366156, A368565, A368566.

c[n_] := PartitionsP[n];
q[n_, k_] := q[n, k] = IntegerPartitions[n][[k]];
r[n_, k_] := r[n, k] = Join[q[n, k], ConstantArray[0, n - Length[q[n, k]]]];
d[u_, v_] := Total[Abs[u - v]];
p[n_] := Flatten[Table[d[r[n, j], r[n, k]] - Abs[j - k], {j, 1, c[n]}, {k, 1, c[n]}]];
Table[Count[p[n], 0], {n, 1, 16}]  (* A368565  *)
Table[Length[Select[p[n], Sign[#] == -1 &]], {n, 1,16}]  (* A368566 *)
Table[Length[Select[p[n], Sign[#] == 1 &]], {n, 1, 16}]  (* A368567 *)

a(n) + A368565(n) + A368566(n) = A001255(n) for n >= 1.

A230025 Triangular array: t(n, k) = number of occurrences of k as the number of outliers in all the partitions of n.

Original entry on oeis.org

1, 0, 2, 1, 0, 2, 1, 2, 0, 2, 1, 2, 2, 0, 2, 1, 2, 4, 2, 0, 2, 1, 4, 2, 4, 2, 0, 2, 2, 2, 6, 2, 6, 2, 0, 2, 2, 6, 2, 8, 2, 6, 2, 0, 2, 2, 4, 12, 2, 8, 2, 8, 2, 0, 2, 2, 8, 6, 14, 2, 10, 2, 8, 2, 0, 2, 3, 6, 14, 8, 18, 2, 10, 2, 10, 2, 0, 2, 3, 10, 10, 20, 10

Offset: 1

Clark Kimberling, Feb 23 2014

Definitions: the self-conjugate portion of a partition p is the portion of the Ferrers graph of p that remains unchanged when p is reflected about its principal diagonal.  The outliers of p are the nodes of the Ferrers graph that lie outside the self-conjugate portion of p.
Sum of numbers in row n is A000041(n).
t(n,k) is the number of partitions p of n such that d(p,p*) = k, where d is the distance function introduced in A366156 and p* is the conjugate of p. - Clark Kimberling, Oct 03 2023

			The first 9 rows:
  1
  0 2
  1 0 2
  1 2 0 2
  1 2 2 0 2
  1 2 4 2 0 2
  1 4 2 4 2 0 2
  2 2 6 2 6 2 0 2
  2 6 2 8 2 6 2 0 2
The Ferrers graph of the partition p = [4,4,1,1] of 10 follows:
  1 1 1 1
  1 1 1 1
  1
  1
The self-conjugate portion of p is
  1 1 1 1
  1 1
  1
  1
so that the number of outliers of p is 2.

Cf. A000041, A366156.

ferrersMatrix[list_] := PadRight[Map[Table[1, {#}] &, #], {#, #} &[Max[#, Length[#]]]] &[list]; conjugatePartition[part_] := Table[Count[#, ?(# >= i &)], {i, First[#]}] &[part]; selfConjugatePortion[list] := ferrersMatrix[#]*ferrersMatrix[conjugatePartition[#]] &[list]; outliers[list_] := Count[Flatten[ferrersMatrix[#] - selfConjugatePortion[#] &[list]], 1]; a[n_] := Map[outliers, IntegerPartitions[n]]; t = Table[Count[a[n], k], {n, 1, 13}, {k, 0, n - 1}]
u = Flatten[t]
(* Peter J. C. Moses, Feb 21 2014 *)

A366461 a(n) = number of partitions of n that have the maximum number of neighbors; see Comments.

Original entry on oeis.org

1, 2, 1, 2, 1, 1, 2, 1, 6, 1, 2, 1, 6, 2, 1, 2, 1, 6, 2, 8, 1, 2, 1, 6, 2, 8, 1, 1, 2, 1, 6, 2, 8, 1, 6, 1, 2, 1, 6, 2, 8, 1, 6, 22, 1, 2, 1, 6, 2, 8, 1, 6, 22, 2, 1, 2, 1, 6, 2, 8, 1, 6, 22, 2, 8, 1, 2, 1, 6, 2, 8, 1, 6, 22, 2, 8, 30, 1, 2, 1, 6, 2, 8, 1, 6, 22

Offset: 1

Clark Kimberling, Oct 12 2023

nonn

Partitions p and q of n are neighbors if d(p,q) = 2, where d is the distance function in A366156.

			Refer to the Example in A366429 to see that a(5) = 1.

Cf. A000041, A366156, A366429.

c[n_] := PartitionsP[n];
q[n_, k_] := q[n, k] = IntegerPartitions[n][[k]];
r[n_, k_] := r[n, k] = Join[q[n, k], ConstantArray[0, n - Length[q[n, k]]]];
d[u_, v_] := d[u, v] = Total[Abs[u - v]];
s[n_, k_] := s[n, k] = Select[Range[c[n]], d[r[n, k], r[n, #]] == 2 &]
t[n_] := t[n] = Table[s[n, k], {k, 1, c[n]}]
a[n_] := Max[Map[Length, t[n]]]
b[n_] := b[n] = Select[t[n], Length[#] == a[n] &]
e[n_] := Length[b[n]]
Table[e[n], {n, 1, 24}]

More terms from Pontus von Brömssen, Oct 24 2023

A366597 Irregular triangular array, read by rows: T(n,k) = number of vertices (partitions) that have degree k in the distance graph of the partitions of n, for k = 1..A366429(n).

Original entry on oeis.org

0, 2, 2, 1, 2, 1, 2, 2, 0, 4, 1, 2, 2, 2, 4, 0, 1, 2, 0, 4, 7, 0, 0, 2, 2, 2, 2, 9, 2, 0, 4, 1, 2, 1, 4, 11, 2, 0, 4, 6, 2, 2, 2, 14, 6, 0, 4, 9, 2, 0, 0, 1, 2, 0, 4, 17, 6, 0, 2, 19, 4, 0, 0, 0, 2, 2, 4, 2, 16, 10, 1, 6, 17, 14, 0, 0, 0, 4, 1, 2, 0, 4, 23

Offset: 1

Clark Kimberling, Oct 16 2023

The distance graph of the partitions of n is defined in A366156.

			First fourteen rows:
   1
   2
   2   1
   2   1   2
   2   0   4   1
   2   2   2   4    0    1
   2   0   4   7    0    0   2
   2   2   2   9    2    0   4   1
   2   1   4   11   2    0   4   6
   2   2   2   14   6    0   4   9   2   0   0   1
   2   0   4   17   6    0   2  19   4   0   0   0   2
   2   4   2   16   10   1   6  17   14  0   0   0   4   1
   2   0   4   23   10   0   2  27   22  1   0   0   4   6
   2   2   2   22   18   2   4  27   32  4   0   0   6   12   2
Enumerate the 7 partitions (vertices) of 5 as follows:
  1: 5
  2: 4,1
  3: 3,2
  4: 3,1,1
  5: 2,2,1
  6: 2,1,1,1
  7: 1,1,1,1,1
Call q a neighbor of p if d(p,q)=2, where d is the distance function in A366156.
The set of neighbors for vertex k, for k = 1..7, is given by
  vertex 1: {2}
  vertex 2: {1,3,4}
  vertex 3: {2,4,5}
  vertex 4: {2,3,5,6}
  vertex 5: {3,4,6}
  vertex 6: {4,5,7}
  vertex 7: {6}
The number of vertices having degrees 1,2,3,4 are 2,0,4,1, respectively, so that row 5 is 2 0 4 1.

Cf. A000041 (row sums), A366429 (row lengths), A366598 (row maxima).

c[n_] := PartitionsP[n]; q[n_, k_] := q[n, k] = IntegerPartitions[n][[k]];
r[n_, k_] := r[n, k] = Join[q[n, k], ConstantArray[0, n - Length[q[n, k]]]];
d[u_, v_] := Total[Abs[u - v]];
s[n_, k_] := Select[Range[c[n]], d[r[n, k], r[n, #]] == 2 &];
s1[n_] := s1[n] = Table[s[n, k], {k, 1, c[n]}];
m[n_] := m[n] = Map[Length, s1[n]];
m1[n_] := m1[n] = Max[m[n]];  (* A366429 *)
t1 = Join[{1}, Table[Count[m[n], i], {n, 2, 15}, {i, 1, m1[n]}]]
Column[t1]
Flatten[t1]

A366598 a(n) = greatest number of vertices having the same degree in the distance graph of the partitions of n.

Original entry on oeis.org

1, 2, 2, 2, 4, 4, 7, 9, 11, 14, 19, 17, 27, 32, 50, 62, 82, 94, 132, 138, 176, 198, 238, 288, 368

Offset: 1

Clark Kimberling, Oct 25 2023

The distance graph of the partitions of n is defined in A366156.

			Enumerate the 7 partitions (= vertices) of 5 as follows:
  1: 5
  2: 4,1
  3: 3,2
  4: 3,1,1
  5: 2,2,1
  6: 2,1,1,1
  7: 1,1,1,1,1
Call q a neighbor of p if d(p,q)=2.
The set of neighbors for vertex k, for k = 1..7, is given by
  vertex 1: {2}
  vertex 2: {1,3,4}
  vertex 3: {2,4,5}
  vertex 4: {2,3,5,6}
  vertex 5: {3,4,6}
  vertex 6: {4,5,7}
  vertex 7: {6}
The number of vertices having degrees 1,2,3,4 are 2,0,4,1, respectively; the greatest of these is 4, so that a(5) = 4.

Cf. A000041, A366156, A366429.

c[n_] := PartitionsP[n]; q[n_, k_] := q[n, k] = IntegerPartitions[n][[k]];
r[n_, k_] := r[n, k] = Join[q[n, k], ConstantArray[0, n - Length[q[n, k]]]];
d[u_, v_] := Total[Abs[u - v]];
s[n_, k_] := Select[Range[c[n]], d[r[n, k], r[n, #]] == 2 &]
s1[n_] := s1[n] = Table[s[n, k], {k, 1, c[n]}]
m[n_] := m[n] = Map[Length, s1[n]]
m1[n_] := m1[n] = Max[m[n]];  (* A366429 *)
t1 = Join[{1}, Table[Count[m[n], i], {n, 2, 25}, {i, 1, m1[n]}]]
Map[Max, t1]

cp's OEIS Frontend

A366745 Triangular array, read by rows: T(n,k) = number of partitions p of n such that d(p, p') = 2k, where p' = contraconjugate of p, and d is the distance function defined in A366156.

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A000097 Number of partitions of n if there are two kinds of 1's and two kinds of 2's.

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A366429 a(n) = maximum degree of vertices in the distance graph of the partitions of n.

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A368565 a(n) = number of pairs (p,q) of partitions of n such that d(p,q) < o(p,q), where d and o are distance functions; see Comments.

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A368566 a(n) = number of pairs (p,q) of partitions of n such that d(p,q) > o(p,q), where d and o are distance functions; see Comments.

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A368564 a(n) = number of pairs (p,q) of partitions of n such that d(p,q) = o(p,q), where d and o are distance functions; see Comments.

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A230025 Triangular array: t(n, k) = number of occurrences of k as the number of outliers in all the partitions of n.

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A366461 a(n) = number of partitions of n that have the maximum number of neighbors; see Comments.

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