A257990 -id:A257990 - cp's OEIS frontend

A000094 Number of trees of diameter 4.

0, 0, 0, 0, 1, 2, 5, 8, 14, 21, 32, 45, 65, 88, 121, 161, 215, 280, 367, 471, 607, 771, 980, 1232, 1551, 1933, 2410, 2983, 3690, 4536, 5574, 6811, 8317, 10110, 12276, 14848, 17941, 21600, 25977, 31146, 37298, 44542, 53132, 63218, 75131, 89089

Offset: 1

N. J. A. Sloane

nonn

Number of partitions of n-1 with at least two parts of size 2 or larger. - Franklin T. Adams-Watters, Jan 13 2006
Also equal to the number of partitions p of n-1 such that max(p)-min(p) > 1. Example: a(7)=5 because we have [5,1],[4,2],[4,1,1],[3,2,1] and [3,1,1,1]. - Giovanni Resta, Feb 06 2006
Also number of partitions of n-1 with at least two parts that are smaller than the largest part. Example: a(7)=5 because we have [4,1,1],[3,2,1],[3,1,1,1],[2,2,1,1,1] and [2,1,1,1,1]. - Emeric Deutsch, May 01 2006
Also number of regions of n-1 that do not contain 1 as a part, n >= 2 (cf. A186114, A206437). - Omar E. Pol, Dec 01 2011
Also rank of the last region of n-1 multiplied by -1, n >= 2 (cf. A194447). - Omar E. Pol, Feb 11 2012
Also sum of ranks of the regions of n-1 that contain emergent parts, n >= 2 (cf. A182699). For the definition of "regions of n" see A206437. - Omar E. Pol, Feb 21 2012

			From _Gus Wiseman_, Apr 12 2019: (Start)
The a(5) = 1 through a(9) = 14 partitions of n-1 with at least two parts of size 2 or larger, or non-hooks, are the following. The Heinz numbers of these partitions are given by A105441.
  (22)  (32)   (33)    (43)     (44)
        (221)  (42)    (52)     (53)
               (222)   (322)    (62)
               (321)   (331)    (332)
               (2211)  (421)    (422)
                       (2221)   (431)
                       (3211)   (521)
                       (22111)  (2222)
                                (3221)
                                (3311)
                                (4211)
                                (22211)
                                (32111)
                                (221111)
The a(5) = 1 through a(9) = 14 partitions of n-1 whose maximum part minus minimum part is at least 2 are the following. The Heinz numbers of these partitions are given by A307516.
  (31)  (41)   (42)    (52)     (53)
        (311)  (51)    (61)     (62)
               (321)   (331)    (71)
               (411)   (421)    (422)
               (3111)  (511)    (431)
                       (3211)   (521)
                       (4111)   (611)
                       (31111)  (3221)
                                (3311)
                                (4211)
                                (5111)
                                (32111)
                                (41111)
                                (311111)
The a(5) = 1 through a(9) = 14 partitions of n-1 with at least two parts that are smaller than the largest part are the following. The Heinz numbers of these partitions are given by A307517.
  (211)  (311)   (321)    (322)     (422)
         (2111)  (411)    (421)     (431)
                 (2211)   (511)     (521)
                 (3111)   (3211)    (611)
                 (21111)  (4111)    (3221)
                          (22111)   (3311)
                          (31111)   (4211)
                          (211111)  (5111)
                                    (22211)
                                    (32111)
                                    (41111)
                                    (221111)
                                    (311111)
                                    (2111111)
(End)

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).

Christian G. Bower, Table of n, a(n) for n = 1..500
J. Riordan, Enumeration of trees by height and diameter, IBM J. Res. Dev. 4 (1960), 473-478.
J. Riordan, The enumeration of trees by height and diameter, IBM Journal 4 (1960), 473-478. (Annotated scanned copy)
Miloslav Znojil, Perturbation theory near degenerate exceptional points, arXiv:2008.00479 [math-ph], 2020.
Index entries for sequences related to trees

Cf. A000041, A206437, A034853, A000147 (diameter 5).

Cf. A006918, A083751, A084835, A105441, A115720, A257990, A307516, A307517, A325164.

g:=x/product(1-x^j,j=1..70)-x-x^2/(1-x)^2: gser:=series(g,x=0,48): seq(coeff(gser,x,n),n=1..46); # Emeric Deutsch, May 01 2006
A000094 := proc(n)
    combinat[numbpart](n-1)-n+1 ;
end proc: # R. J. Mathar, May 17 2016

t=Table[PartitionsP[n]-n,{n,0,45}];
ReplacePart[t,0,1]
(* Clark Kimberling, Mar 05 2012 *)
CoefficientList[1/QPochhammer[x]-x/(1-x)^2-1+O[x]^50, x] (* Jean-François Alcover, Feb 04 2016 *)

a(n+1) = A000041(n)-n for n>0. - John W. Layman
G.f.: x/product(1-x^j,j=1..infinity)-x-x^2/(1-x)^2. - Emeric Deutsch, May 01 2006
G.f.: sum(sum(x^(i+j+1)/product(1-x^k, k=i..j), i=1..j-2), j=3..infinity). - Emeric Deutsch, May 01 2006
a(n+1) = Sum_{m=1..n} A083751(m). - Gregory Gerard Wojnar, Oct 13 2020

More terms from Franklin T. Adams-Watters, Jan 13 2006

A325233 Heinz numbers of integer partitions with Dyson rank 1.

Original entry on oeis.org

3, 10, 15, 25, 28, 42, 63, 70, 88, 98, 105, 132, 147, 175, 198, 208, 220, 245, 297, 308, 312, 330, 343, 462, 468, 484, 495, 520, 544, 550, 693, 702, 726, 728, 770, 780, 816, 825, 1053, 1078, 1089, 1092, 1144, 1155, 1170, 1210, 1216, 1224, 1300, 1352, 1360

Offset: 1

Gus Wiseman, Apr 13 2019

nonn

Numbers whose maximum prime index is one greater than their number of prime indices counted with multiplicity.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The sequence of terms together with their prime indices begins:
     3: {2}
    10: {1,3}
    15: {2,3}
    25: {3,3}
    28: {1,1,4}
    42: {1,2,4}
    63: {2,2,4}
    70: {1,3,4}
    88: {1,1,1,5}
    98: {1,4,4}
   105: {2,3,4}
   132: {1,1,2,5}
   147: {2,4,4}
   175: {3,3,4}
   198: {1,2,2,5}
   208: {1,1,1,1,6}
   220: {1,1,3,5}
   245: {3,4,4}
   297: {2,2,2,5}
   308: {1,1,4,5}

FindStat, St000145: The Dyson rank of a partition

Positions of 1's in A257541.

Cf. A001222, A047993, A056239, A061395, A063995, A101198, A106529, A112798, A257990, A263297, A325225, A325234, A325235.

Select[Range[1000],PrimePi[FactorInteger[#][[-1,1]]]-PrimeOmega[#]==1&]

A352517 Number of weak excedances (parts on or above the diagonal) of the n-th composition in standard order.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 2, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 2, 2, 2, 2, 3, 2, 2

Offset: 0

Gus Wiseman, Mar 23 2022

nonn

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions. See also A000120, A059893, A070939, A114994, A225620.

			The 169th composition in standard order is (2,2,3,1), with weak excedances {1,2,3}, so a(169) = 3.

MathOverflow, Why 'excedances' of permutations? [closed].

Positive positions of first appearances are A164894.
The version for partitions is A257990.
The strong opposite version is A352514, counted by A352521 (first A219282).
The opposite version is A352515, counted by A352522 (first column A238874).
The strong version is A352516, counted by A352524 (first column A008930).
The triangle A352525 counts these compositions (first column A177510).
A008292 is the triangle of Eulerian numbers (version without zeros).
A011782 counts compositions.
A173018 counts permutations by number of excedances, weak A123125.
A238349 counts comps by fixed points, first col A238351, rank stat A352512.
A352489 is the weak excedance set of A122111.
A352523 counts comps by unfixed points, first A010054, rank stat A352513.
Cf. A088218, A098825, A114088, A115994, A131577, A238352, A352513, A352523.

stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
pdw[y_]:=Length[Select[Range[Length[y]],#<=y[[#]]&]];
Table[pdw[stc[n]],{n,0,30}]

A325163 Heinz number of the inner lining partition of the integer partition with Heinz number n.

Original entry on oeis.org

1, 2, 3, 3, 5, 5, 7, 5, 10, 7, 11, 7, 13, 11, 14, 7, 17, 14, 19, 11, 22, 13, 23, 11, 21, 17, 21, 13, 29, 22, 31, 11, 26, 19, 33, 22, 37, 23, 34, 13, 41, 26, 43, 17, 33, 29, 47, 13, 55, 33, 38, 19, 53, 33, 39, 17, 46, 31, 59, 26, 61, 37, 39, 13, 51, 34, 67, 23

Offset: 1

Gus Wiseman, Apr 05 2019

nonn

The k-th part of the inner lining partition of an integer partition is the number of squares in its Young diagram that are k diagonal steps from the lower-right boundary. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The partition with Heinz number 7865 is (6,5,5,3), with diagram
  o o o o o o
  o o o o o
  o o o o o
  o o o
which has diagonal distances
  3 3 3 2 1 1
  3 2 2 2 1
  2 2 1 1 1
  1 1 1
so the inner lining partition is (9,6,4), which has Heinz number 2093, so a(7865) = 2093.

Cf. A052126, A056239, A064989, A065770, A093641, A112798, A188674, A252464, A257990, A297113, A325133, A325135, A325164, A325167, A325169.

Table[Times@@Prime/@(-Differences[Total/@Take[FixedPointList[If[#=={},{},DeleteCases[Rest[#]-1,0]]&,Reverse[Flatten[Cases[If[n==1,{},FactorInteger[n]],{p_,k_}:>Table[PrimePi[p],{k}]]]]],{1,-2}]]),{n,100}]

A325166 Size of the internal portion of the integer partition with Heinz number n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 2, 0, 0, 2, 0, 1, 2, 1, 0, 1, 2, 1, 2, 1, 0, 3, 0, 0, 2, 1, 3, 2, 0, 1, 2, 1, 0, 3, 0, 1, 3, 1, 0, 1, 3, 3, 2, 1, 0, 3, 3, 1, 2, 1, 0, 3, 0, 1, 3, 0, 3, 3, 0, 1, 2, 4, 0, 2, 0, 1, 4, 1, 4, 3, 0, 1, 3, 1, 0, 3, 3, 1, 2, 1, 0, 4, 4, 1, 2, 1, 3, 1, 0, 4, 3, 3, 0, 3, 0, 1, 5

Offset: 1

Gus Wiseman, Apr 05 2019

nonn

The internal portion of an integer partition consists of all squares in the Young diagram that have a square both directly below and directly to the right.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The partition with Heinz number 7865 is (6,5,5,3), with diagram
  o o o o o o
  o o o o o
  o o o o o
  o o o
with internal portion
  o o o o o
  o o o o
  o o o
of size 12, so a(7865) = 12.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Positions of zeros are A174090.

Cf. A001221, A001222, A052126, A056239, A061395, A064989, A065770, A112798, A252464, A257990, A297113, A325133, A325135, A325167, A325169.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[If[n==1,0,Total[primeMS[n]]-Max[primeMS[n]]-Length[primeMS[n]]+Length[Union[primeMS[n]]]],{n,100}]

A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i,2] * primepi(f[i,1]))); }
A061395(n) = if(1==n, 0, primepi(vecmax(factor(n)[, 1])));
A325166(n) = (A056239(n) - A061395(n) - bigomega(n) + omega(n)); \\ Antti Karttunen, Apr 14 2019

a(n) = A056239(n) - A061395(n) - A001222(n) + A001221(n).

a(n) = A056239(n) - A297113(n).

More terms from Antti Karttunen, Apr 14 2019

A325178 Difference between the length of the minimal square containing and the maximal square contained in the Young diagram of the integer partition with Heinz number n.

Original entry on oeis.org

0, 0, 1, 1, 2, 1, 3, 2, 0, 2, 4, 2, 5, 3, 1, 3, 6, 1, 7, 2, 2, 4, 8, 3, 1, 5, 1, 3, 9, 1, 10, 4, 3, 6, 2, 2, 11, 7, 4, 3, 12, 2, 13, 4, 1, 8, 14, 4, 2, 1, 5, 5, 15, 2, 3, 3, 6, 9, 16, 2, 17, 10, 2, 5, 4, 3, 18, 6, 7, 2, 19, 3, 20, 11, 1, 7, 3, 4, 21, 4, 2, 12

Offset: 1

Gus Wiseman, Apr 08 2019

nonn

The maximal square contained in the Young diagram of an integer partition is called its Durfee square, and its length is the rank of the partition.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The partition (3,3,2,1) has Heinz number 150 and diagram
  o o o
  o o o
  o o
  o
containing maximal square
  o o
  o o
and contained in minimal square
  o o o o
  o o o o
  o o o o
  o o o o
so a(150) = 4 - 2 = 2.

Richard P. Stanley, Enumerative Combinatorics, Volume 2, Cambridge University Press, 1999, p. 289.

Wikipedia, Durfee square.

Positions of zeros are A062457. Positions of 1's are A325179. Positions of 2's are A325180.

Cf. A001222, A046660, A051924, A056239, A061395, A093641, A096771, A115994, A243055, A257990, A263297, A325192, A325195.

durf[n_]:=Length[Select[Range[PrimeOmega[n]],Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]][[#]]>=#&]];
codurf[n_]:=If[n==1,0,Max[PrimeOmega[n],PrimePi[FactorInteger[n][[-1,1]]]]];
Table[codurf[n]-durf[n],{n,100}]

a(n) = A263297(n) - A257990(n).

A352829 Number of strict integer partitions y of n with a fixed point y(i) = i.

Original entry on oeis.org

0, 1, 0, 0, 0, 1, 2, 2, 2, 2, 2, 2, 3, 4, 6, 8, 10, 12, 14, 16, 18, 20, 23, 26, 30, 36, 42, 50, 60, 70, 82, 96, 110, 126, 144, 163, 184, 208, 234, 264, 298, 336, 380, 430, 486, 550, 622, 702, 792, 892, 1002, 1125, 1260, 1408, 1572, 1752, 1950, 2168, 2408, 2672

Offset: 0

Gus Wiseman, May 15 2022

nonn

			The a(11) = 2 through a(17) = 12 partitions (A-F = 10..15):
  (92)   (A2)   (B2)    (C2)    (D2)     (E2)     (F2)
  (821)  (543)  (643)   (653)   (753)    (763)    (863)
         (921)  (A21)   (743)   (843)    (853)    (953)
                (5431)  (B21)   (C21)    (943)    (A43)
                        (5432)  (6432)   (D21)    (E21)
                        (6431)  (6531)   (6532)   (7532)
                                (7431)   (7432)   (7631)
                                (54321)  (7531)   (8432)
                                         (8431)   (8531)
                                         (64321)  (9431)
                                                  (65321)
                                                  (74321)

The non-strict version is A001522 (unproved, ranked by A352827 or A352874).
The version for permutations is A002467, complement A000166.
The reverse version is A096765 (or A025147 shifted right once).
The non-strict reverse version is A238395, ranked by A352872.
The complement is counted by A352828, non-strict A064428 (unproved, ranked by A352826 or A352873).
The version for compositions is A352875, complement A238351.
A000041 counts partitions, strict A000009.
A000700 counts self-conjugate partitions, ranked by A088902.
A008290 counts permutations by fixed points, unfixed A098825.
A115720 and A115994 count partitions by their Durfee square.
A238349 counts compositions by fixed points, complement A352523.
A238352 counts reversed partitions by fixed points, rank statistic A352822.
A238394 counts reversed partitions without a fixed point, ranked by A352830.
A352833 counts partitions by fixed points.
Cf. A008292, A064410, A111133, A114088, A118199, A188674, A257990, A352823, A352824, A352825, A352832.

pq[y_]:=Length[Select[Range[Length[y]],#==y[[#]]&]];
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&pq[#]>0&]],{n,0,30}]

G.f.: Sum_{n>=1} q^(n*(3*n-1)/2)*Product_{k=1..n-1} (1+q^k)/(1-q^k). - Jeremy Lovejoy, Sep 26 2022

A325192 Regular triangle read by rows where T(n,k) is the number of integer partitions of n such that the difference between the length of the minimal square containing and the maximal square contained in the Young diagram is k.

Original entry on oeis.org

1, 1, 0, 0, 2, 0, 0, 1, 2, 0, 1, 0, 2, 2, 0, 0, 2, 1, 2, 2, 0, 0, 3, 2, 2, 2, 2, 0, 0, 2, 4, 3, 2, 2, 2, 0, 0, 1, 7, 4, 4, 2, 2, 2, 0, 1, 0, 6, 8, 5, 4, 2, 2, 2, 0, 0, 2, 5, 11, 8, 6, 4, 2, 2, 2, 0, 0, 3, 4, 12, 12, 9, 6, 4, 2, 2, 2, 0, 0, 4, 5, 13, 17, 12, 10, 6, 4, 2, 2, 2, 0

Offset: 0

Gus Wiseman, Apr 08 2019

The maximal square contained in the Young diagram of an integer partition is called its Durfee square, and its length is the rank of the partition.

			Triangle begins:
  1
  1  0
  0  2  0
  0  1  2  0
  1  0  2  2  0
  0  2  1  2  2  0
  0  3  2  2  2  2  0
  0  2  4  3  2  2  2  0
  0  1  7  4  4  2  2  2  0
  1  0  6  8  5  4  2  2  2  0
  0  2  5 11  8  6  4  2  2  2  0
  0  3  4 12 12  9  6  4  2  2  2  0
  0  4  5 13 17 12 10  6  4  2  2  2  0
  0  3  9 12 20 18 13 10  6  4  2  2  2  0
  0  2 12 15 23 25 18 14 10  6  4  2  2  2  0
  0  1 15 19 26 30 26 19 14 10  6  4  2  2  2  0
Row 9 counts the following partitions (empty columns not shown):
   333   432    54      63       72        711       81         9
         441    522     621      6111      3111111   21111111   111111111
         3222   531     51111    411111
         3321   5211    222111   2211111
         4221   22221   321111
         4311   32211
                33111
                42111

Richard P. Stanley, Enumerative Combinatorics, Volume 2, Cambridge University Press, 1999, p. 289.

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
Wikipedia, Durfee square.

Row sums are A000041. Column k = 1 is A325181. Column k = 2 is A325182.
Cf. A096771, A257990, A263297, A325178, A325179, A325180, A325200.
Cf. A115995, A368985.

durf[ptn_]:=Length[Select[Range[Length[ptn]],ptn[[#]]>=#&]];
codurf[ptn_]:=Max[Length[ptn],Max[ptn]];
Table[Length[Select[IntegerPartitions[n],codurf[#]-durf[#]==k&]],{n,0,15},{k,0,n}]

row(n)={my(r=vector(n+1)); if(n==0, r[1]=1, forpart(p=n, my(c=1); while(c<#p && cAndrew Howroyd, Jan 12 2024

Sum_{k=1..n} k*T(n,k) = A368985(n) - A115995(n). - Andrew Howroyd, Jan 12 2024

A352831 Numbers whose weakly increasing prime indices y have exactly one fixed point y(i) = i.

Original entry on oeis.org

2, 4, 8, 9, 10, 12, 14, 16, 22, 24, 26, 27, 28, 32, 34, 36, 38, 40, 44, 46, 48, 52, 58, 60, 62, 63, 64, 68, 70, 72, 74, 75, 76, 80, 81, 82, 86, 88, 92, 94, 96, 98, 99, 104, 106, 108, 110, 112, 116, 117, 118, 120, 122, 124, 125, 128, 130, 132, 134, 135, 136

Offset: 1

Gus Wiseman, Apr 08 2022

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The terms together with their prime indices begin:
      2: {1}             36: {1,1,2,2}         74: {1,12}
      4: {1,1}           38: {1,8}             75: {2,3,3}
      8: {1,1,1}         40: {1,1,1,3}         76: {1,1,8}
      9: {2,2}           44: {1,1,5}           80: {1,1,1,1,3}
     10: {1,3}           46: {1,9}             81: {2,2,2,2}
     12: {1,1,2}         48: {1,1,1,1,2}       82: {1,13}
     14: {1,4}           52: {1,1,6}           86: {1,14}
     16: {1,1,1,1}       58: {1,10}            88: {1,1,1,5}
     22: {1,5}           60: {1,1,2,3}         92: {1,1,9}
     24: {1,1,1,2}       62: {1,11}            94: {1,15}
     26: {1,6}           63: {2,2,4}           96: {1,1,1,1,1,2}
     27: {2,2,2}         64: {1,1,1,1,1,1}     98: {1,4,4}
     28: {1,1,4}         68: {1,1,7}           99: {2,2,5}
     32: {1,1,1,1,1}     70: {1,3,4}          104: {1,1,1,6}
     34: {1,7}           72: {1,1,1,2,2}      106: {1,16}
For example, 63 is in the sequence because its prime indices {2,2,4} have a unique fixed point at the second position.

* = unproved
These are the positions of 1's in A352822.
*The reverse version for no fixed points is A352826, counted by A064428.
*The reverse version is A352827, counted by A001522 (strict A352829).
The version for no fixed points is A352830, counted by A238394.
These partitions are counted by A352832, compositions A240736.
Allowing more than one fixed point gives A352872, counted by A238395.
A000700 counts self-conjugate partitions, ranked by A088902.
A001222 counts prime indices, distinct A001221.
A008290 counts permutations by fixed points, nonfixed A098825.
A056239 adds up prime indices, row sums of A112798 and A296150.
A115720 and A115994 count partitions by their Durfee square.
A238349 counts compositions by fixed points, complement A352523.
A238352 counts reversed partitions by fixed points.
A352833 counts partitions by fixed points.
Cf. A062457, A064410, A065770, A177510, A257990, A342192, A349158, A351983, A352520, A352823, A352824.

pq[y_]:=Length[Select[Range[Length[y]],#==y[[#]]&]];
Select[Range[100],pq[Flatten[Cases[FactorInteger[#],{p_,k_}:>Table[PrimePi[p],{k}]]]]==1&]

A118199 Number of partitions of n having no parts equal to the size of their Durfee squares.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 2, 3, 5, 7, 10, 13, 18, 23, 31, 40, 53, 68, 89, 113, 146, 184, 234, 293, 369, 458, 572, 706, 874, 1073, 1320, 1611, 1970, 2393, 2909, 3518, 4255, 5122, 6167, 7394, 8862, 10585, 12637, 15038, 17886, 21213, 25141, 29723, 35112, 41383, 48737, 57278

Offset: 0

Emeric Deutsch, Apr 14 2006

nonn

a(n) = A118198(n,0).
From Gus Wiseman, May 21 2022: (Start)
Also the number of integer partitions of n > 0 that have a fixed point but whose conjugate does not, ranked by A353316. For example, the a(5) = 1 through a(10) = 10 partitions are:
  11111   222      322       422        522         622
          111111   2221      2222       3222        4222
                   1111111   3221       4221        5221
                             22211      22221       22222
                             11111111   32211       32221
                                        222111      42211
                                        111111111   222211
                                                    322111
                                                    2221111
                                                    1111111111
Partitions w/ a fixed point: A001522 (unproved), ranked by A352827 (cf. A352874).
Partitions w/o a fixed point: A064428 (unproved), ranked by A352826 (cf. A352873).
Partitions w/ a fixed point and a conjugate fixed point: A188674, reverse A325187, ranked by A353317.
Partitions w/o a fixed point or conjugate fixed point: A188674 (shifted).
(End)

			a(7) = 3 because we have [7] with size of Durfee square 1, [4,3] with size of Durfee square 2 and [3,3,1] with size of Durfee square 2.

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000 (terms 0..1000 from Alois P. Heinz)

Column k=0 of A118198.
A000041 counts partitions, strict A000009.
A000700 = self-conjugate partitions, ranked by A088902, complement A330644.
A002467 counts permutations with a fixed point, complement A000166.
A064410 counts partitions of crank 0, ranked by A342192.
A115720 and A115994 count partitions by Durfee square, rank stat A257990.
A238352 counts reversed partitions by fixed points, rank statistic A352822.
A238394 counts reversed partitions without a fixed point, ranked by A352830.
A238395 counts reversed partitions with a fixed point, ranked by A352872.
A352833 counts partitions by fixed points.
Cf. A114088, A300788, A325039, A350839, A352828, A352829, A352832.

g:=1+sum(x^(k^2+k)/(1-x^k)/product((1-x^i)^2,i=1..k-1),k=1..20): gser:=series(g,x=0,60): seq(coeff(gser,x,n),n=0..54);
# second Maple program::
b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, b(n-i, i))))
    end:
a:= n-> add(add(b(k, d) *b(n-d*(d+1)-k, d-1),
                k=0..n-d*(d+1)), d=0..floor(sqrt(n))):
seq(a(n), n=0..70);  # Alois P. Heinz, Apr 09 2012

b[n_, i_] :=  b[n, i] = If[n == 0, 1, If[i<1, 0, b[n, i-1] + If[i>n, 0, b[n-i, i]]]]; a[n_] := Sum[Sum[b[k, d]*b[n-d*(d+1)-k, d-1], {k, 0, n-d*(d+1)}], {d, 0, Floor[Sqrt[n]]}]; Table[a[n], {n, 0, 70}] (* Jean-François Alcover, May 22 2015, after Alois P. Heinz *)
pq[y_]:=Length[Select[Range[Length[y]],#==y[[#]]&]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Table[Length[Select[IntegerPartitions[n],pq[#]>0&&pq[conj[#]]==0&]],{n,0,30}] (* a(0) = 0, Gus Wiseman, May 21 2022 *)

G.f.: 1+sum(x^(k^2+k)/[(1-x^k)*product((1-x^i)^2, i=1..k-1)], k=1..infinity).

a(n) ~ exp(Pi*sqrt(2*n/3)) / (16*n*sqrt(3)). - Vaclav Kotesovec, Jun 12 2025

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