A352486 -id:A352486 - cp's OEIS frontend

A088902 Numbers n such that n = product (p_k)^(c_k) and set of its (c_k k's)'s is a self-conjugate partition, where p_k is k-th prime and c_k > 0.

1, 2, 6, 9, 20, 30, 56, 75, 84, 125, 176, 210, 264, 350, 416, 441, 624, 660, 735, 1088, 1100, 1386, 1560, 1632, 1715, 2310, 2401, 2432, 2600, 3267, 3276, 3648, 4080, 5390, 5445, 5460, 5888, 6800, 7546, 7722, 8568, 8832, 9120, 12705, 12740, 12870, 13689

Offset: 1

Naohiro Nomoto, Nov 28 2003

The Heinz numbers of the self-conjugate partitions. We define the Heinz number of a partition p = [p_1, p_2, ..., p_r] to be Product(p_j-th prime, j=1..r) (a concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). For example, for the partition [1, 1, 1, 4] we get 2*2*2*7 = 56. It is in the sequence since [1,1,1,4] is self-conjugate. - Emeric Deutsch, Jun 05 2015

			20 is in the sequence because 20 = 2^2 * 5^1 = (p_1)^2 *(p_3)^1, (two 1's, one 3's) = (1,1,3) is a self-conjugate partition of 5.
From _Gus Wiseman_, Jun 28 2022: (Start)
The terms together with their prime indices begin:
    1: ()
    2: (1)
    6: (2,1)
    9: (2,2)
   20: (3,1,1)
   30: (3,2,1)
   56: (4,1,1,1)
   75: (3,3,2)
   84: (4,2,1,1)
  125: (3,3,3)
  176: (5,1,1,1,1)
  210: (4,3,2,1)
  264: (5,2,1,1,1)
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..400

Fixed points of A122111.
Cf. A242422, A215366.
A002110 (primorial numbers) is a subsequence.
After a(1) and a(2), a subsequence of A241913.
These partitions are counted by A000700.
The same count comes from A258116.
The complement is A352486, counted by A330644.
These are the positions of zeros in A352491.
A000041 counts integer partitions, strict A000009.
A325039 counts partitions w/ product = conjugate product, ranked by A325040.
Heinz number (rank) and partition:
- A003963 = product of partition, conjugate A329382.
- A008480 = number of permutations of partition, conjugate A321648.
- A056239 = sum of partition.
- A296150 = parts of partition, reverse A112798, conjugate A321649.
- A352487 = less than conjugate, counted by A000701.
- A352488 = greater than or equal to conjugate, counted by A046682.
- A352489 = less than or equal to conjugate, counted by A046682.
- A352490 = greater than conjugate, counted by A000701.
Cf. A000720, A098825, A195017, A238351, A238745, A347450, A350841.

with(numtheory): c := proc (n) local B, C: B := proc (n) local pf: pf := op(2, ifactors(n)): [seq(seq(pi(op(1, op(i, pf))), j = 1 .. op(2, op(i, pf))), i = 1 .. nops(pf))] end proc: C := proc (P) local a: a := proc (j) local c, i: c := 0: for i to nops(P) do if j <= P[i] then c := c+1 else end if end do: c end proc: [seq(a(k), k = 1 .. max(P))] end proc: mul(ithprime(C(B(n))[q]), q = 1 .. nops(C(B(n)))) end proc: SC := {}: for i to 14000 do if c(i) = i then SC := `union`(SC, {i}) else end if end do: SC; # Emeric Deutsch, May 09 2015

Select[Range[14000], Function[n, n == If[n == 1, 1, Module[{l = #, m = 0}, Times @@ Power @@@ Table[l -= m; l = DeleteCases[l, 0]; {Prime@ Length@ l, m = Min@ l}, Length@ Union@ l]] &@ Catenate[ConstantArray[PrimePi@ #1, #2] & @@@ FactorInteger@ n]]]] (* Michael De Vlieger, Aug 27 2016, after JungHwan Min at A122111 *)

More terms from David Wasserman, Aug 26 2005

A000701 One half of number of non-self-conjugate partitions; also half of number of asymmetric Ferrers graphs with n nodes.

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 5, 7, 10, 14, 20, 27, 37, 49, 66, 86, 113, 146, 190, 242, 310, 392, 497, 623, 782, 973, 1212, 1498, 1851, 2274, 2793, 3411, 4163, 5059, 6142, 7427, 8972, 10801, 12989, 15572, 18646, 22267, 26561, 31602, 37556, 44533, 52743, 62338, 73593

Offset: 0

N. J. A. Sloane

Also number of cycle types of odd permutations.
Also number of partitions of n with an odd number of even parts. There is no restriction on the odd parts. - N. Sato, Jul 20 2005. E.g., a(6)=5 because we have [6],[4,1,1],[3,2,1],[2,2,2] and [2,1,1,1,1]. - Emeric Deutsch, Mar 02 2006
Also number of partitions of n with largest part not congruent to n modulo 2: a(2*n)=A027193(2*n), a(2*n+1)=A027187(2*n+1); a(n)=A000041(n)-A046682(n). - Reinhard Zumkeller, Apr 22 2006
From Gus Wiseman, Mar 31 2022: (Start)
Also the number of integer partitions of n with Heinz number greater than that of their conjugate, where the Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). These partitions are ranked by A352490. The complement is counted by A046682. For example, the a(n) partitions for n = 2...8 are:
  (11)  (111)  (211)   (221)    (222)     (331)      (2222)
               (1111)  (2111)   (2211)    (2221)     (3221)
                       (11111)  (3111)    (3211)     (3311)
                                (21111)   (22111)    (22211)
                                (111111)  (31111)    (32111)
                                          (211111)   (41111)
                                          (1111111)  (221111)
                                                     (311111)
                                                     (2111111)
                                                     (11111111)
Also the number of integer partitions of n with Heinz number less than that of their conjugate, ranked by A352487. For example, the a(n) partitions for n = 2...8 are:
  (2)  (3)  (4)   (5)   (6)    (7)    (8)
            (31)  (32)  (33)   (43)   (44)
                  (41)  (42)   (52)   (53)
                        (51)   (61)   (62)
                        (411)  (322)  (71)
                               (421)  (422)
                               (511)  (431)
                                      (521)
                                      (611)
                                      (5111)
(End)

			G.f. = x^2 + x^3 + 2*x^4 + 3*x^5 + 5*x^6 + 7*x^7 + 10*x^8 + 14*x^9 + ...

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).

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
Brian Hopkins and Michael A. Jones, Shift-Induced Dynamical Systems on Partitions and Compositions, Electron. J. Combin. 13 (2006), Research Paper 80, see p. 10.
M. Osima, On the irreducible representations of the symmetric group, Canad. J. Math., 4 (1952), 381-384.

Cf. A000041, A027187, A027193, A046682, A118302, A236559, A236914.
A000700 counts self-conjugate partitions, ranked by A088902.
A330644 counts non-self-conjugate partitions, ranked by A352486.
Heinz number (rank) and partition:
- A122111 = rank of conjugate.
- A296150 = parts of partition, conjugate A321649.
- A352487 = rank less than conjugate, counted by A000701.
- A352488 = rank greater than or equal to conjugate, counted by A046682.
- A352489 = rank less than or equal to conjugate, counted by A046682.
- A352490 = rank greater than conjugate, counted by A000701.
- A352491 = rank minus conjugate.
Cf. A114088, A115994, A171966, A238352, A258116, A325039.

with(combinat); A000701 := n->(numbpart(n)-A000700(n))/2;

a41 = PartitionsP; a700[n_] := SeriesCoefficient[ Product[1 + x^k, {k, 1, n, 2}], {x, 0, n}]; a[0] = 0; a[n_] := (a41[n] - a700[n])/2; Table[a[n], {n, 0, 48}] (* Jean-François Alcover, Feb 21 2012, after first formula *)
a[ n_] := SeriesCoefficient[ (1 / QPochhammer[ x] - 1 / QPochhammer[ x, -x]) / 2, {x, 0, n}]; (* Michael Somos, Aug 25 2015 *)
a[ n_] := SeriesCoefficient[ (1 - EllipticTheta[ 4, 0, x^2]) / (2 QPochhammer[ x]), {x, 0, n}]; (* Michael Somos, Aug 25 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x] Sum[ x^(2 k) / QPochhammer[ x^2, x^2, k], {k, 1, n/2, 2}], {x, 0, n}] (* Michael Somos, Aug 25 2015 *)
a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Sum[ (1 / QPochhammer[ x, x, k]^2 - 1 / QPochhammer[ x^2, x^2, k]) x^k^2, {k, Sqrt@n}] / 2, {x, 0, n}]]; (* Michael Somos, Aug 25 2015 *)
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Table[Length[Select[IntegerPartitions[n],Times@@Prime/@#>Times@@Prime/@conj[#]&]],{n,0,15}] (* Gus Wiseman, Mar 31 2022 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (1 - eta(x^2 + A)^2 / eta(x^4 + A) ) / (2 * eta(x + A)), n))}; /* Michael Somos, Aug 25 2015 */

q='q+O('q^60); concat([0, 0], Vec((1-eta(q^2)^2/eta(q^4))/(2*eta(q)))) \\ Altug Alkan, Sep 26 2018

a(n) = (A000041(n) - A000700(n))/2.
From Bill Gosper, Aug 08 2005: (Start)
Sum a(n) q^n = q^2 + q^3 + 2 q^4 + 3 q^5 + 5 q^6 + 7 q^7 + ...
= -( Sum_{n>=1} (-q^2)^(n^2) ) / ( Sum_{ n = -oo..oo } (-1)^n q^(n(3n-1)/2) )
= (- q; q){oo} Sum{n>=1} q^(2(2n-1))/(q^2;q^2)_{2n-1}
= (1/(q;q)_oo - 1/(q;-q)_oo)/2
= (1/(q;q)_oo - (-q;q^2)_oo)/2
= Sum{k>=0} ( 1/((q;q)_k)^2 - 1/(q^2;q^2)_k ) q^(k^2)/2
using the "q-Pochhammer" notation (a;q)n := Product{k=0..n-1} (1 - a*q^k).
(End)
a(n) = p(n-2) - p(n-8) + p(n-18) - p(n-32) + ... + (-1)^(k+1)*p(n-2*k^2) + ..., where p() is A000041(). E.g., a(20) = p(18) - p(12) + p(2) = 385 - 77 + 2 = 310. - Vladeta Jovovic, Aug 08 2004
G.f.: (1/2)*(1 - Product_{j>=1} (1-x^(2j))/(1+x^(2j)))/Product_{j>=1} (1 - x^j). - Emeric Deutsch, Mar 02 2006
a(2*n) = A236559(n). a(2*n + 1) = A236914(n). - Michael Somos, Aug 25 2015
a(n) = A330644(n)/2. - Omar E. Pol, Jan 10 2020
a(n) = A000041(n) - A046682(n) = A046682(n) - A000700(n). - Gus Wiseman, Mar 31 2022

Better description and more terms from Christian G. Bower, Apr 27 2000

A046682 Number of cycle types of conjugacy classes of all even permutations of n elements.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 6, 8, 12, 16, 22, 29, 40, 52, 69, 90, 118, 151, 195, 248, 317, 400, 505, 632, 793, 985, 1224, 1512, 1867, 2291, 2811, 3431, 4186, 5084, 6168, 7456, 9005, 10836, 13026, 15613, 18692, 22316, 26613, 31659, 37619, 44601, 52815, 62416, 73680, 86809, 102162

Offset: 0

Vladeta Jovovic

Also number of partitions of n with even number of even parts. There is no restriction on the odd parts.
a(n) = u(n) + v(n), n >= 2, of the Osima reference, p. 383.
Also number of partitions of n with largest part congruent to n modulo 2: a(2*n) = A027187(2*n), a(2*n-1) = A027193(2*n-1); a(n) = A000041(n) - A000701(n). - Reinhard Zumkeller, Apr 22 2006
Equivalently, number of partitions of n with number of parts having the same parity as n. - Olivier Gérard, Apr 04 2012
Also number of distinct free Young diagrams (Ferrers graphs with n nodes). Free Young diagrams are distinct when none is a rigid transformation (translation, rotation, reflection or glide reflection) of another. - Jani Melik, May 08 2016
Let the cycle type of an even permutation be represented by the partition A=(O1,O2,...,Oi,E1,E2,...,E2j), where the Os are parts with odd length and the Es are parts with even lengths, and where j may be zero, using Reinhard Zumkeller's observation that the partition associated with a cycle type of an even permutation has an even number of even parts. The set of even cycle types enumerated here can be considered a monoid under a binary operation *: Let A be as above and B=(o1,o2,...,ok,e1,e2,...,e2m). A*B is the partition (O1o1,O1o2,...,O1ok,O1e1,...,O1e2m,O2o1,...,O2e2m,...,Oio1,...,Oie2m,E1o1,...,E1e2m,...,E2je2m). This product has 2im+2jk+4jm even parts, so it represents the cycle type of an even permutation. - Richard Locke Peterson, Aug 20 2018
From Gus Wiseman, Mar 31 2022: (Start)
Also the number of integer partitions of n with Heinz number greater than or equal to that of their conjugate, where the Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). These partitions are ranked by A352488. The complement is counted by A000701. For example, the a(n) partitions for n = 1...7 are:
  (1)  (11)  (21)   (22)    (221)    (222)     (331)
             (111)  (211)   (311)    (321)     (2221)
                    (1111)  (2111)   (2211)    (3211)
                            (11111)  (3111)    (4111)
                                     (21111)   (22111)
                                     (111111)  (31111)
                                               (211111)
                                               (1111111)
Also the number of integer partitions of n with Heinz number less than or equal to their conjugate, ranked by A352489. For example, the a(n) partitions for n = 1...7 are:
  (1)  (2)  (3)   (4)   (5)    (6)    (7)
            (21)  (22)  (32)   (33)   (43)
                  (31)  (41)   (42)   (52)
                        (311)  (51)   (61)
                               (321)  (322)
                               (411)  (421)
                                      (511)
                                      (4111)
(End)

			1 + x + x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 6*x^6 + 8*x^7 + 12*x^8 + 16*x^9 + ...
a(3)=2 since cycle types of even permutations of 3 elements is (.)(.)(.), (...).
a(4)=3 since cycle types of even permutations of 4 elements is (.)(.)(.)(.), (...)(.), (..)(..).
a(5)=4 (free Young diagrams):
  XXXXX XXXX. XXX.. XXX..
  ..... X.... XX... X....
  ..... ..... ..... X....
  ..... ..... ..... .....
  ..... ..... ..... .....

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
George E. Andrews, David Newman, The Minimal Excludant in Integer Partitions, J. Int. Seq., Vol. 23 (2020), Article 20.2.3.
J. Huh and B. Kim, The number of equivalence classes arising from partition involutions, Int. J. Number Theory, 16 (2020), 925-939.
M. Osima, On the irreducible representations of the symmetric group, Canad. J. Math., 4 (1952), 381-384.
Sheila Sundaram, On a positivity conjecture in the character table of S_n, arXiv:1808.01416 [math.CO], 2018.

Cf. A000701, A006950, A015128.
For the number of conjugacy classes of the alternating group A_n, n>=2, see A000702.
Cf. A118301.
A000041 counts integer partitions.
A000700 counts self-conjugate partitions, ranked by A088902.
A330644 counts non-self-conjugate partitions, ranked by A352486.
Heinz number (rank) and partition:
- A122111 = rank of conjugate.
- A296150 = parts of partition, conjugate A321649.
- A352487 = rank less than conjugate, counted by A000701.
- A352488 = rank greater than or equal to conjugate, counted by A046682.
- A352489 = rank less than or equal to conjugate, counted by A046682.
- A352490 = rank greater than conjugate, counted by A000701.
- A352491 = rank minus conjugate.
Cf. A114088, A115994, A171966, A238352, A258116, A321648, A325039.

seq(add((-1)^(n-k)*combinat:-numbpart(n,k),k=0..n),n=0..48); # Peter Luschny, Aug 03 2015

max = 48; f[q_] := Sum[(-q^2)^n^2, {n, 0, max}]/Product[1-q^n, {n, 1, max}]; CoefficientList[ Series[f[q], {q, 0, max}], q] (* Jean-François Alcover, Oct 18 2011, after g.f. *)
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Table[Length[Select[IntegerPartitions[n],Times@@Prime/@#>=Times@@Prime/@conj[#]&]],{n,0,15}] (* Gus Wiseman, Mar 31 2022 *)

list(lim)=my(q='q);Vec(sum(n=0,sqrt(lim),(-q^2)^(n^2))/prod(n=1,lim,1-q^n)+O(q^(lim\1+1))) \\ Charles R Greathouse IV, Oct 18 2011

{a(n) = if( n<0, 0, (numbpart(n) + polcoeff( 1 / prod( k=1, n, 1 + (-x)^k, 1 + x * O(x^n)), n)) / 2)} /* Michael Somos, Jul 24 2012 */

G.f.: Sum_{n>=0} (-q^2)^(n^2) / Product_{m>=1} (1-q^m ) = ( 1/Product_{m>=1} (1-q^m) + Product_{m>=1} (1+q^(2*m-1) ) ) / 2. - Mamuka Jibladze, Sep 07 2003
a(n) = (A000041(n) + A000700(n)) / 2.
a(n) = A000041(n) - A000701(n). - Gus Wiseman, Mar 31 2022

A352822 Number of fixed points y(i) = i, where y is the weakly increasing sequence of prime indices of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Apr 05 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 prime indices of 6500 are {1,1,3,3,3,6} with fixed points at positions {1,3,6}, so a(6500) = 3.

* = unproved
Positions of first appearances are A002110.
The triangle version is A238352.
Positions of 0's are A352830, counted by A238394.
Positions of 1's are A352831, counted by A352832.
A version for compositions is A352512, complement A352513, triangle A238349.
The complement is A352823.
The reverse version is A352824, complement A352825.
A000700 counts self-conjugate partitions, ranked by A088902.
A001222 counts prime indices, distinct A001221.
*A001522 counts partitions with a fixed point, ranked by A352827.
A056239 adds up prime indices, row sums of A112798 and A296150.
*A064428 counts partitions without a fixed point, ranked by A352826.
A122111 represents partition conjugation using Heinz numbers.
A124010 gives prime signature, sorted A118914, conjugate rank A238745.
A115720 and A115994 count partitions by their Durfee square.
A238395 counts reversed partitions with a fixed point, ranked by A352872.
Cf. A065770, A093641, A098825, A114088, A118199, A257990, A325163, A325164, A325169, A342192, A352486-A352491, A352829.

f:= proc(n) local F,J,t;
  F:= sort(ifactors(n)[2],(s,t) -> s[1] numtheory:-pi(t[1])$t[2], F);
  nops(select(t -> J[t]=t, [$1..nops(J)]));
end proc:
map(f, [$1..200]); # Robert Israel, Apr 11 2023

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

A352822(n) = { my(f=factor(n),i=0,c=0); for(k=1,#f~,while(f[k,2], f[k,2]--; i++; c += (i==primepi(f[k,1])))); (c); }; \\ Antti Karttunen, Apr 11 2022

a(n) = A001222(n) - A352823(n). - Antti Karttunen, Apr 11 2022

Data section extended up to 105 terms by Antti Karttunen, Apr 11 2022

A352827 Heinz numbers of integer partitions y with a fixed point y(i) = i. Such a fixed point is unique if it exists.

Original entry on oeis.org

2, 4, 8, 9, 15, 16, 18, 21, 27, 30, 32, 33, 36, 39, 42, 45, 51, 54, 57, 60, 63, 64, 66, 69, 72, 78, 81, 84, 87, 90, 93, 99, 102, 108, 111, 114, 117, 120, 123, 125, 126, 128, 129, 132, 135, 138, 141, 144, 153, 156, 159, 162, 168, 171, 174, 175, 177, 180, 183

Offset: 1

Gus Wiseman, Apr 06 2022

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The terms together with their prime indices begin:
    2: (1)
    4: (1,1)
    8: (1,1,1)
    9: (2,2)
   15: (3,2)
   16: (1,1,1,1)
   18: (2,2,1)
   21: (4,2)
   27: (2,2,2)
   30: (3,2,1)
   32: (1,1,1,1,1)
   33: (5,2)
   36: (2,2,1,1)
   39: (6,2)
   42: (4,2,1)
   45: (3,2,2)
   51: (7,2)
   54: (2,2,2,1)
For example, the partition (3,2,2) with Heinz number 45 has a fixed point at position 2, so 45 is in the sequence.

* = unproved
*These partitions are counted by A001522, strict A352829.
*The complement is A352826, counted by A064428.
The complement reverse version is A352830, counted by A238394.
The reverse version is A352872, counted by A238395
A000700 counts self-conjugate partitions, ranked by A088902.
A001222 counts prime indices, distinct A001221.
A008290 counts permutations by fixed points, unfixed A098825.
A056239 adds up prime indices, row sums of A112798 and A296150.
A115720 and A115994 count partitions by their Durfee square.
A122111 represents partition conjugation using Heinz numbers.
A124010 gives prime signature, sorted A118914, conjugate rank A238745.
A238349 counts compositions by fixed points, complement A352523.
A238352 counts reversed partitions by fixed points, rank statistic A352822.
A352828 counts strict partitions without a fixed point.
A352833 counts partitions by fixed points.
Cf. A062457, A064410, A065770, A093641, A257990, A342192, A352486, A352823, A352824 (characteristic function), A352825, A352831.

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

A352523 Number of integer compositions of n with exactly k nonfixed points (parts not on the diagonal).

Original entry on oeis.org

1, 1, 0, 0, 2, 0, 1, 1, 2, 0, 0, 4, 2, 2, 0, 0, 5, 5, 4, 2, 0, 1, 3, 12, 8, 6, 2, 0, 0, 7, 14, 19, 14, 8, 2, 0, 0, 8, 21, 33, 32, 22, 10, 2, 0, 0, 9, 30, 54, 63, 54, 32, 12, 2, 0, 1, 6, 47, 80, 116, 116, 86, 44, 14, 2, 0, 0, 11, 53, 129, 194, 229, 202, 130, 58, 16, 2, 0

Offset: 0

Gus Wiseman, Mar 26 2022

A nonfixed point in a composition c is an index i such that c_i != i.

			Triangle begins:
   1
   1   0
   0   2   0
   1   1   2   0
   0   4   2   2   0
   0   5   5   4   2   0
   1   3  12   8   6   2   0
   0   7  14  19  14   8   2   0
   0   8  21  33  32  22  10   2   0
   0   9  30  54  63  54  32  12   2   0
   1   6  47  80 116 116  86  44  14   2   0
   ...
For example, row n = 6 counts the following compositions (empty column indicated by dot):
  (123)  (6)   (24)    (231)    (2112)   (21111)    .
         (15)  (33)    (312)    (2121)   (111111)
         (42)  (51)    (411)    (3111)
               (114)   (1113)   (11112)
               (132)   (1122)   (11121)
               (141)   (1311)   (11211)
               (213)   (2211)
               (222)   (12111)
               (321)
               (1131)
               (1212)
               (1221)

John Tyler Rascoe, Rows n = 0..130, flattened

Column k = 0 is A010054.
Row sums are A011782.
The version for permutations is A098825.
The corresponding rank statistic is A352513.
Column k = 1 is A352520.
A238349 and A238350 count comps by fixed points, first col A238351, rank stat A352512.
A352486 gives the nonfixed points of A122111, counted by A330644.
A352521 counts comps by strong nonexcedances, first A219282, stat A352514.
A352522 counts comps by weak nonexcedances, first col A238874, stat A352515.
A352524 counts comps by strong excedances, first col A008930, stat A352516.
A352525 counts comps by weak excedances, first col A177510, stat A352517.
Cf. A000700, A064428, A088218, A114088, A115994, A123125, A173018, A238352.

b:= proc(n, i) option remember; expand(`if`(n=0, 1,
      add(`if`(i=j, 1, x)*b(n-j, i+1), j=1..n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, 1)):
seq(T(n), n=0..12);  # Alois P. Heinz, Mar 19 2025

pnq[y_]:=Length[Select[Range[Length[y]],#!=y[[#]]&]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],pnq[#]==k&]],{n,0,9},{k,0,n}]

T_xy(max_row) = {my(N=max_row+1, x='x+O('x^N), h= sum(i=0, N, prod(j=1, i, y*(x/(1-x)-x^j)+x^j))); vector(N, n, my(r=Vecrev(polcoeff(h, n-1))); if(n<2, r, concat(r,[0])))}
T_xy(10) \\ John Tyler Rascoe, Mar 21 2025

G.f.: Sum_{i>=0} Product_{j=1..i} y*(x/(1-x) - x^j) + x^j. - John Tyler Rascoe, Mar 19 2025

A352512 Number of fixed points in the n-th composition in standard order.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Mar 26 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.

A fixed point of composition c is an index i such that c_i = i.

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

The version counting permutations is A008290, unfixed A098825.
The triangular version is A238349, first column A238351.
Unfixed points are counted by A352513, triangle A352523, first A352520.
A011782 counts compositions.
A088902 gives the fixed points of A122111, counted by A000700.
A352521 counts comps by strong nonexcedances, first A219282, stat A352514.
A352522 counts comps by weak nonexcedances, first col A238874, stat A352515.
A352524 counts comps by strong excedances, first col A008930, stat A352516.
A352525 counts comps by weak excedances, first col A177510, stat A352517.
Cf. A088218, A114088, A115994, A238352, A330644, A350841, A352486.

stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
pq[y_]:=Length[Select[Range[Length[y]],#==y[[#]]&]];
Table[pq[stc[n]],{n,0,100}]

A000120(n) = A352512(n) + A352513(n).

A352826 Heinz numbers of integer partitions y without a fixed point y(i) = i. Such a fixed point is unique if it exists.

Original entry on oeis.org

1, 3, 5, 6, 7, 10, 11, 12, 13, 14, 17, 19, 20, 22, 23, 24, 25, 26, 28, 29, 31, 34, 35, 37, 38, 40, 41, 43, 44, 46, 47, 48, 49, 50, 52, 53, 55, 56, 58, 59, 61, 62, 65, 67, 68, 70, 71, 73, 74, 75, 76, 77, 79, 80, 82, 83, 85, 86, 88, 89, 91, 92, 94, 95, 96, 97

Offset: 1

Gus Wiseman, Apr 06 2022

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The terms together with their prime indices begin:
      1: ()          24: (2,1,1,1)     47: (15)
      3: (2)         25: (3,3)         48: (2,1,1,1,1)
      5: (3)         26: (6,1)         49: (4,4)
      6: (2,1)       28: (4,1,1)       50: (3,3,1)
      7: (4)         29: (10)          52: (6,1,1)
     10: (3,1)       31: (11)          53: (16)
     11: (5)         34: (7,1)         55: (5,3)
     12: (2,1,1)     35: (4,3)         56: (4,1,1,1)
     13: (6)         37: (12)          58: (10,1)
     14: (4,1)       38: (8,1)         59: (17)
     17: (7)         40: (3,1,1,1)     61: (18)
     19: (8)         41: (13)          62: (11,1)
     20: (3,1,1)     43: (14)          65: (6,3)
     22: (5,1)       44: (5,1,1)       67: (19)
     23: (9)         46: (9,1)         68: (7,1,1)

* = unproved
*These partitions are counted by A064428, strict A352828.
The complement is A352827.
The reverse version is A352830, counted by A238394.
A000700 counts self-conjugate partitions, ranked by A088902.
A001222 counts prime indices, distinct A001221.
*A001522 counts partitions with a fixed point.
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.
A122111 represents partition conjugation using Heinz numbers.
A124010 gives prime signature, sorted A118914.
A238349 counts compositions by fixed points, complement A352523.
A238352 counts reversed partitions by fixed points, rank statistic A352822.
A238395 counts reversed partitions with a fixed point, ranked by A352872.
A352833 counts partitions by fixed points.
Cf. A062457, A064410, A065770, A093641, A257990, A342192, A352486, A352823, A352824, A352825, A352831, A352832.

pq[y_]:=Length[Select[Range[Length[y]],#==y[[#]]&]];
Select[Range[100],pq[Reverse[Flatten[Cases[FactorInteger[#],{p_,k_}:>Table[PrimePi[p],{k}]]]]]==0&]

A352513 Number of nonfixed points in the n-th composition in standard order.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Mar 27 2022

nonn

A nonfixed point in a composition c is an index i such that c_i != i.

			The 169th composition in standard order is (2,2,3,1), with nonfixed points {1,4}, so a(169) = 2.

The version counting permutations is A098825, fixed A008290.
Fixed points are counted by A352512, triangle A238349, first A238351.
The triangular version is A352523, first nontrivial column A352520.
A011782 counts compositions.
A352486 gives the nonfixed points of A122111, counted by A330644.
A352521 counts comps by strong nonexcedances, first A219282, stat A352514.
A352522 counts comps by weak nonexcedances, first col A238874, stat A352515.
A352524 counts comps by strong excedances, first col A008930, stat A352516.
A352525 counts comps by weak excedances, first col A177510, stat A352517.
Cf. A000700, A088218, A088902, A114088, A115994, A238352, A350841.

stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
pnq[y_]:=Length[Select[Range[Length[y]],#!=y[[#]]&]];
Table[pnq[stc[n]],{n,0,100}]

A000120(n) = A352512(n) + A352513(n).

A352830 Numbers whose weakly increasing prime indices y have no fixed points y(i) = i.

Original entry on oeis.org

1, 3, 5, 7, 11, 13, 15, 17, 19, 21, 23, 25, 29, 31, 33, 35, 37, 39, 41, 43, 47, 49, 51, 53, 55, 57, 59, 61, 65, 67, 69, 71, 73, 77, 79, 83, 85, 87, 89, 91, 93, 95, 97, 101, 103, 105, 107, 109, 111, 113, 115, 119, 121, 123, 127, 129, 131, 133, 137, 139, 141

Offset: 1

Gus Wiseman, Apr 06 2022

nonn

First differs from A325128 in lacking 75.
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.
All terms are odd.

			The terms together with their prime indices begin:
      1: {}        35: {3,4}     69: {2,9}     105: {2,3,4}
      3: {2}       37: {12}      71: {20}      107: {28}
      5: {3}       39: {2,6}     73: {21}      109: {29}
      7: {4}       41: {13}      77: {4,5}     111: {2,12}
     11: {5}       43: {14}      79: {22}      113: {30}
     13: {6}       47: {15}      83: {23}      115: {3,9}
     15: {2,3}     49: {4,4}     85: {3,7}     119: {4,7}
     17: {7}       51: {2,7}     87: {2,10}    121: {5,5}
     19: {8}       53: {16}      89: {24}      123: {2,13}
     21: {2,4}     55: {3,5}     91: {4,6}     127: {31}
     23: {9}       57: {2,8}     93: {2,11}    129: {2,14}
     25: {3,3}     59: {17}      95: {3,8}     131: {32}
     29: {10}      61: {18}      97: {25}      133: {4,8}
     31: {11}      65: {3,6}    101: {26}      137: {33}
     33: {2,5}     67: {19}     103: {27}      139: {34}

* = unproved
These partitions are counted by A238394, strict A025147.
These are the zeros of A352822.
*The reverse version is A352826, counted by A064428 (strict A352828).
*The complement reverse version is A352827, counted by A001522.
The complement is 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.
A114088 counts partitions by excedances.
A115720 and A115994 count partitions by their Durfee square.
A122111 represents partition conjugation using Heinz numbers.
A124010 gives prime signature, sorted A118914, conjugate rank A238745.
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, A093641, A257990, A342192, A352486, A352823, A352824, A352825, A352831.

pq[y_]:=Length[Select[Range[Length[y]],#==y[[#]]&]];
Select[Range[100],pq[Flatten[Cases[FactorInteger[#],{p_,k_}:>Table[PrimePi[p],{k}]]]]==0&]

cp's OEIS Frontend

A088902 Numbers n such that n = product (p_k)^(c_k) and set of its (c_k k's)'s is a self-conjugate partition, where p_k is k-th prime and c_k > 0.

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A000701 One half of number of non-self-conjugate partitions; also half of number of asymmetric Ferrers graphs with n nodes.

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A046682 Number of cycle types of conjugacy classes of all even permutations of n elements.

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A352822 Number of fixed points y(i) = i, where y is the weakly increasing sequence of prime indices of n.

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A352827 Heinz numbers of integer partitions y with a fixed point y(i) = i. Such a fixed point is unique if it exists.

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A352523 Number of integer compositions of n with exactly k nonfixed points (parts not on the diagonal).

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A352512 Number of fixed points in the n-th composition in standard order.