A098825 -id:A098825 - cp's OEIS frontend

A008290 Triangle T(n,k) of rencontres numbers (number of permutations of n elements with k fixed points).

1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 9, 8, 6, 0, 1, 44, 45, 20, 10, 0, 1, 265, 264, 135, 40, 15, 0, 1, 1854, 1855, 924, 315, 70, 21, 0, 1, 14833, 14832, 7420, 2464, 630, 112, 28, 0, 1, 133496, 133497, 66744, 22260, 5544, 1134, 168, 36, 0, 1, 1334961, 1334960, 667485, 222480, 55650, 11088, 1890, 240, 45, 0, 1

Offset: 0

N. J. A. Sloane

This is a binomial convolution triangle (Sheffer triangle) of the Appell type: (exp(-x)/(1-x),x), i.e., the e.g.f. of column k is (exp(-x)/(1-x))*(x^k/k!). See the e.g.f. given by V. Jovovic below. - Wolfdieter Lang, Jan 21 2008
The formula T(n,k) = binomial(n,k)*A000166(n-k), with the derangements numbers (subfactorials) A000166 (see also the Charalambides reference) shows the Appell type of this triangle. - Wolfdieter Lang, Jan 21 2008
T(n,k) is the number of permutations of {1,2,...,n} having k pairs of consecutive right-to-left minima (0 is considered a right-to-left minimum for each permutation). Example: T(4,2)=6 because we have 1243, 1423, 4123, 1324, 3124 and 2134; for example, 1324 has right-to-left minima in positions 0-1,3-4 and 2134 has right-to-left minima in positions 0,2-3-4, the consecutive ones being joined by "-". - Emeric Deutsch, Mar 29 2008
T is an example of the group of matrices outlined in the table in A132382--the associated matrix for the sequence aC(0,1). - Tom Copeland, Sep 10 2008
A refinement of this triangle is given by A036039. - Tom Copeland, Nov 06 2012
This triangle equals (A211229(2*n,2*k)) n,k >= 0. - Peter Bala, Dec 17 2014

			exp((y-1)*x)/(1-x) = 1 + y*x + (1/2!)*(1+y^2)*x^2 + (1/3!)*(2 + 3*y + y^3)*x^3 + (1/4!)*(9 + 8*y + 6*y^2 + y^4)*x^4 + (1/5!)*(44 + 45*y + 20*y^2 + 10*y^3 + y^5)*x^5 + ...
Triangle begins:
       1
       0      1
       1      0     1
       2      3     0     1
       9      8     6     0    1
      44     45    20    10    0    1
     265    264   135    40   15    0   1
    1854   1855   924   315   70   21   0  1
   14833  14832  7420  2464  630  112  28  0 1
  133496 133497 66744 22260 5544 1134 168 36 0 1
...
From _Peter Bala_, Feb 13 2017: (Start)
The infinitesimal generator has integer entries given by binomial(n,k)*(n-k-1)! for n >= 2 and 0 <= k <= n-2 and begins
   0
   0  0
   1  0  0
   2  3  0  0
   6  8  6  0 0
  24 30 20 10 0 0
...
It is essentially A238363 (unsigned and omitting the main diagonal), A211603 (with different offset) and appears to be A092271, again without the main diagonal. (End)

Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002, p. 173, Table 5.2 (without row n=0 and column k=0).
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 194.
Arnold Kaufmann, Introduction à la combinatorique en vue des applications, Dunod, Paris, 1968. See p. 92.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 65.

Alois P. Heinz, Rows n = 0..150, flattened (first 51 rows from T. D. Noe)
Taha Akbari, Prove using combinatorics Sum_{k=0..n} (k-1)^2 D_n(k)=n!, Mathematics Stack Exchange, Jun 06 2017
Paul Barry, General Eulerian Polynomials as Moments Using Exponential Riordan Arrays, Journal of Integer Sequences, 16 (2013), #13.9.6.
Stefano Capparelli, Margherita Maria Ferrari, Emanuele Munarini, and Norma Zagaglia Salvi, A Generalization of the "Problème des Rencontres", J. Int. Seq. 21 (2018), #18.2.8.
Bhadrachalam Chitturi and Krishnaveni K S, Adjacencies in Permutations, arXiv preprint arXiv:1601.04469 [cs.DM], 2016. See Table 1.
S. K. Das and N. Deo, Rencontres graphs: a family of bipartite graphs, Fib. Quart., Vol. 25, No. 3, August 1987, 250-262.
Robert W. Donley Jr, Binomial arrays and generalized Vandermonde identities, arXiv:1905.01525 [math.CO], 2019.
FindStat - Combinatorial Statistic Finder, The number of adjacencies of a permutation, 0 appended, The number of fixed points of a permutation
I. Kaplansky, Symbolic solution of certain problems in permutations, Bull. Amer. Math. Soc., 50 (1944), 906-914.
J. Liese and J. Remmel, Q-analogues of the number of permutations with k-excedances, PU. M. A. Vol. 21 (2010), No. 2, pp. 285-320 (see E_{n,0}(x) in Table 1 p. 291).
L. Takacs, On the "problème des ménages", Discr. Math. 36 (3) (1981) 289-297, Table 2.
Wikipedia, Rencontres numbers.
Index entries for sequences related to permutations with fixed points

Mirror of triangle A098825.
Columns give A000166, A000240, A000387, A000449, A000475, A129135, A129136, A129149, A129153, A129217, A129218, A129238, A129255.
Cf. A055137, A008291, A098558.
Cf. A080955.
Cf. A000012, A000142 (row sums), A000354.
Cf. A170942. Sub-triangle of A211229.
Cf. A092271, A111492, A211603, A238363.
T(2n,n) gives A281262.
Cf. A000023, A133314, A238385, A320582, A335111.

a008290 n k = a008290_tabl !! n !! k
a008290_row n = a008290_tabl !! n
a008290_tabl = map reverse a098825_tabl
-- Reinhard Zumkeller, Dec 16 2013

T:= proc(n,k) T(n, k):= `if`(k=0, `if`(n<2, 1-n, (n-1)*
      (T(n-1, 0)+T(n-2, 0))), binomial(n, k)*T(n-k, 0))
    end:
seq(seq(T(n, k), k=0..n), n=0..12);  # Alois P. Heinz, Mar 15 2013

a[0] = 1; a[1] = 0; a[n_] := Round[n!/E] /; n >= 1 size = 8; Table[Binomial[n, k]a[n - k], {n, 0, size}, {k, 0, n}] // TableForm (* Harlan J. Brothers, Mar 19 2007 *)
T[n_, k_] := Subfactorial[n-k]*Binomial[n, k]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 12 2017 *)
T[n_, k_] := If[n<1, Boole[n==0 && k==0], T[n, k] = T[n-1, k-1] + T[n-1, k]*(n-1-k) + T[n-1, k+1]*(k+1)]; (* Michael Somos, Sep 13 2024 *)
T[0, 0]:=1; T[n_, 0]:=T[n, 0]=n  T[n-1, 0]+(-1)^n; T[n_, k_]:=T[n, k]=n/k T[n-1, k-1];
Flatten@Table[T[n, k], {n, 0, 9}, {k, 0, n}] (* Oliver Seipel, Nov 26 2024 *)

{T(n, k) = if(k<0 || k>n, 0, n!/k! * sum(i=0, n-k, (-1)^i/i!))}; /* Michael Somos, Apr 26 2000 */

T(n, k) = T(n-1, k)*n + binomial(n, k)*(-1)^(n-k) = T(n, k-1)/k + binomial(n, k)*(-1)^(n-k)/(n-k+1) = T(n-1, k-1)*n/k = T(n-k, 0)*binomial(n, k) = A000166(n-k)*binomial(n,k) [with T(0, 0) = 1]; so T(n, n) = 1, T(n, n-1) = 0, T(n, n-2) = n*(n-1)/2 for n >= 0.
Sum_{k=0..n} T(n, k) = Sum_{k=0..n} k * T(n, k) = n! for all n > 0, n, k integers. - Wouter Meeussen, May 29 2001
From Vladeta Jovovic, Aug 12 2002: (Start)
O.g.f. for k-th column: (1/k!)*Sum_{i>=k} i!*x^i/(1+x)^(i+1).
O.g.f. for k-th row: k!*Sum_{i=0..k} (-1)^i/i!*(1-x)^i. (End)
E.g.f.: exp((y-1)*x)/(1-x). - Vladeta Jovovic, Aug 18 2002
E.g.f. for number of permutations with exactly k fixed points is x^k/(k!*exp(x)*(1-x)). - Vladeta Jovovic, Aug 25 2002
Sum_{k=0..n} T(n, k)*x^k is the permanent of the n X n matrix with x's on the diagonal and 1's elsewhere; for x = 0, 1, 2, 3, 4, 5, 6 see A000166, A000142, A000522, A010842, A053486, A053487, A080954. - Philippe Deléham, Dec 12 2003; for x = 1+i see A009551 and A009102. - John M. Campbell, Oct 11 2011
T(n, k) = Sum_{j=0..n} A008290(n, j)*k^(n-j) is the permanent of the n X n matrix with 1's on the diagonal and k's elsewhere; for k = 0, 1, 2 see A000012, A000142, A000354. - Philippe Deléham, Dec 13 2003
T(n,k) = Sum_{j=0..n} (-1)^(j-k)*binomial(j,k)*n!/j!. - Paul Barry, May 25 2006
T(n,k) = (n!/k!)*Sum_{j=0..n-k} ((-1)^j)/j!, 0 <= k <= n. From the Appell type of the triangle and the subfactorial formula.
T(n,0) = n*Sum_{j=0..n-1} (j/(j+1))*T(n-1,j), T(0,0)=1. From the z-sequence of this Sheffer triangle z(j)=j/(j+1) with e.g.f. (1-exp(x)*(1-x))/x. See the W. Lang link under A006232 for Sheffer a- and z-sequences. - Wolfdieter Lang, Jan 21 2008
T(n,k) = (n/k)*T(n-1,k-1) for k >= 1. See above. From the a-sequence of this Sheffer triangle a(0)=1, a(n)=0, n >= 1 with e.g.f. 1. See the W. Lang link under A006232 for Sheffer a- and z-sequences. - Wolfdieter Lang, Jan 21 2008
From Henk P. J. van Wijk, Oct 29 2012: (Start)
T(n,k) =              T(n-1,k)*(n-1-k) + T(n-1,k+1)*(k+1) for k=0 and
T(n,k) = T(n-1,k-1) + T(n-1,k)*(n-1-k) + T(n-1,k+1)*(k+1) for k>=1.
(End)
T(n,k) = A098825(n,n-k). - Reinhard Zumkeller, Dec 16 2013
Sum_{k=0..n} k^2 * T(n, k) = 2*n! if n > 1. - Michael Somos, Jun 06 2017
From Tom Copeland, Jul 26 2017: (Start)
The lowering and raising operators of this Appell sequence of polynomials P(n,x) are L = d/dx and R = x + d/dL log[exp(-L)/(1-L)] = x-1 + 1/(1-L) = x + L + L^2 - ... such that L P(n,x) = n P(n-1,x) and R P(n,x) = P(n+1,x).
P(n,x) = (1-L)^(-1) exp(-L) x^n = (1+L+L^2+...)(x-1)^n = n! Sum_{k=0..n} (x-1)^k / k!.
The formalism of A133314 applies to the pair of entries A008290 and A055137.
The polynomials of this pair P_n(x) and Q_n(x) are umbral compositional inverses; i.e., P_n(Q.(x)) = x^n = Q_n(P.(x)), where, e.g., (Q.(x))^n = Q_n(x).
For more on the infinitesimal generator, noted by Bala below, see A238385. (End)
Sum_{k=0..n} k^m * T(n,k) = A000110(m)*n! if n >= m. - Zhujun Zhang, May 24 2019
Sum_{k=0..n} (k+1) * T(n,k) = A098558(n). - Alois P. Heinz, Mar 11 2022
From Alois P. Heinz, May 20 2023: (Start)
Sum_{k=0..n} (-1)^k * T(n,k) = A000023(n).
Sum_{k=0..n} (-1)^k * k * T(n,k) = A335111(n). (End)
T(n,k) = A145224(n,k)+A145225(n,k), refined by even and odd perms. - R. J. Mathar, Jul 06 2023

Comments and more terms from Michael Somos, Apr 26 2000 and Christian G. Bower, Apr 26 2000

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.

Original entry on oeis.org

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

A238349 Triangle T(n,k) read by rows: T(n,k) is the number of compositions of n with k parts p at position p (fixed points), n>=0, 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 1, 1, 0, 2, 1, 1, 0, 3, 4, 1, 0, 0, 6, 7, 3, 0, 0, 0, 11, 16, 4, 1, 0, 0, 0, 22, 29, 12, 1, 0, 0, 0, 0, 42, 60, 23, 3, 0, 0, 0, 0, 0, 82, 120, 47, 7, 0, 0, 0, 0, 0, 0, 161, 238, 100, 12, 1, 0, 0, 0, 0, 0, 0, 316, 479, 198, 30, 1, 0, 0, 0, 0, 0, 0, 0, 624, 956, 404, 61, 3, 0, 0, 0, 0, 0, 0, 0, 0, 1235, 1910, 818, 126, 7, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Joerg Arndt and Alois P. Heinz, Feb 25 2014

T(n*(n+3)/2,n) = A227682(n).
From Vaclav Kotesovec, Sep 07 2014: (Start)
In general, column k is asymptotic to c(k) * 2^n. The constants c(k) numerically:
c(0) = 0.144394047543301210639449860964615390044455952420342... = A048651/2
c(1) = 0.231997216225445223894202367545783700531838988546098... = c(0)*A065442
c(2) = 0.104261929557371534733906196116707679501974368826074...
c(3) = 0.017956317806894073430249112172514186063327165575720...
c(4) = 0.001343254222922697613125145839110293324517874530073...
c(5) = 0.000046459767012163920051487037952792359225887287888...
c(6) = 0.000000768651747857094917953943327540619110335556499...
c(7) = 0.000000006200599904985793344094393321042983316604040...
c(8) = 0.000000000024656652167851516076173236693314090168122...
c(9) = 0.000000000000048633746319332356416193899916110113745...
c(10)= 0.000000000000000047750743608910618576944191079881479...
c(20)= 1.05217230403079700467566...*10^(-63)
For big k is c(k) ~ m * 2^(-k*(k+1)/2), where m = 1/(4*c(0)) = 1/(2*A048651) = 1.7313733097275318...
(End)

			Triangle starts:
00:  1,
01:  0, 1,
02:  1, 1, 0,
03:  2, 1, 1, 0,
04:  3, 4, 1, 0, 0,
05:  6, 7, 3, 0, 0, 0,
06:  11, 16, 4, 1, 0, 0, 0,
07:  22, 29, 12, 1, 0, 0, 0, 0,
08:  42, 60, 23, 3, 0, 0, 0, 0, 0,
09:  82, 120, 47, 7, 0, 0, 0, 0, 0, 0,
10:  161, 238, 100, 12, 1, 0, 0, 0, 0, 0, 0,
11:  316, 479, 198, 30, 1, 0, 0, 0, 0, 0, 0, 0,
12:  624, 956, 404, 61, 3, 0, 0, 0, 0, 0, 0, 0, 0,
13:  1235, 1910, 818, 126, 7, 0, 0, 0, 0, 0, 0, 0, 0, 0,
14:  2449, 3817, 1652, 258, 16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
15:  4864, 7633, 3319, 537, 30, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
...
From _Gus Wiseman_, Apr 03 2022: (Start)
Row n = 5 counts the following compositions (empty columns indicated by dots):
  (5)     (14)     (113)   .  .  .
  (23)    (32)     (122)
  (41)    (131)    (1211)
  (212)   (221)
  (311)   (1112)
  (2111)  (1121)
          (11111)
(End)

M. Archibald, A. Blecher and A. Knopfmacher, Fixed points in compositions and words, accepted by the Journal of Integer Sequences.

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 0..10010 (rows 0..140, flattened)
M. Archibald, A. Blecher, and A. Knopfmacher, Fixed Points in Compositions and Words, J. Int. Seq., Vol. 23 (2020), Article 20.11.1.

Row sums are A011782.
Columns k=0-10 give: A238351, A240736, A240737, A240738, A240739, A240740, A240741, A240742, A240743, A240744, A240745.
The version for permutations is A008290.
The version with all zeros removed is A238350.
The version for reversed partitions is A238352.
The corresponding rank statistic is A352512, nonfixed A352513.
The version for nonfixed points is A352523, A352520 (k=1).
Below: comps = compositions, first = column k=0, stat = rank statistic.
- A352521 counts comps by strong nonexcedances, first A219282, stat A352514.
- A352522 counts comps by weak nonexcedances, first A238874, stat A352515.
- A352524 counts comps by strong excedances, first A008930, stat A352516.
- A352525 counts comps by weak excedances, A177510 (k=1), stat A352517.
Cf. A008292, A088218, A098825, A227682, A352487.

b:= proc(n, i) option remember; `if`(n=0, 1, expand(
       add(b(n-j, i+1)*`if`(i=j, x, 1), j=1..n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, 1)):
seq(T(n), n=0..15);

b[n_, i_] := b[n, i] = If[n == 0, 1, Expand[Sum[b[n-j, i+1]*If[i == j, x, 1], {j, 1, n}]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, 1]]; Table[T[n], {n, 0, 15}] // Flatten (* Jean-François Alcover, Jan 06 2015, after Alois P. Heinz *)
pq[y_]:=Length[Select[Range[Length[y]],#==y[[#]]&]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],pq[#]==k&]],{n,0,9},{k,0,n}] (* Gus Wiseman, Apr 03 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

A177510 Number of compositions (p0, p1, p2, ...) of n with pi - p0 <= i and pi >= p0.

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 14, 25, 46, 87, 167, 324, 634, 1248, 2466, 4887, 9706, 19308, 38455, 76659, 152925, 305232, 609488, 1217429, 2432399, 4860881, 9715511, 19421029, 38826059, 77626471, 155211785, 310357462, 620608652, 1241046343, 2481817484, 4963191718, 9925669171, 19850186856, 39698516655, 79394037319

Offset: 0

Mats Granvik, Dec 11 2010

nonn

a(0)=1, otherwise row sums of A179748.
For n>=1 cumulative sums of A008930.
a(n) is proportional to A048651*A000079. The error (a(n)-A048651*A000079) divided by sequence A186425 tends to the golden ratio A001622. This can be seen when using about 1000 decimals of the constant A048651 = 0.2887880950866024212... - [Mats Granvik, Jan 01 2015]
From Gus Wiseman, Mar 31 2022: (Start)
Also the number of integer compositions of n with exactly one part on or above the diagonal. For example, the a(1) = 1 through a(5) = 8 compositions are:
  (1)  (2)   (3)    (4)     (5)
       (11)  (21)   (31)    (41)
             (111)  (112)   (212)
                    (211)   (311)
                    (1111)  (1112)
                            (1121)
                            (2111)
                            (11111)
(End)

			From _Joerg Arndt_, Mar 24 2014: (Start)
The a(7) = 25 such compositions are:
01:  [ 1 1 1 1 1 1 1 ]
02:  [ 1 1 1 1 1 2 ]
03:  [ 1 1 1 1 2 1 ]
04:  [ 1 1 1 1 3 ]
05:  [ 1 1 1 2 1 1 ]
06:  [ 1 1 1 2 2 ]
07:  [ 1 1 1 3 1 ]
08:  [ 1 1 1 4 ]
09:  [ 1 1 2 1 1 1 ]
10:  [ 1 1 2 1 2 ]
11:  [ 1 1 2 2 1 ]
12:  [ 1 1 2 3 ]
13:  [ 1 1 3 1 1 ]
14:  [ 1 1 3 2 ]
15:  [ 1 2 1 1 1 1 ]
16:  [ 1 2 1 1 2 ]
17:  [ 1 2 1 2 1 ]
18:  [ 1 2 1 3 ]
19:  [ 1 2 2 1 1 ]
20:  [ 1 2 2 2 ]
21:  [ 1 2 3 1 ]
22:  [ 2 2 3 ]
23:  [ 2 3 2 ]
24:  [ 3 4 ]
25:  [ 7 ]
(End)

Cf. A238859 (compositions with subdiagonal growth), A238876 (partitions with subdiagonal growth), A001227 (partitions into distinct parts with subdiagonal growth).
Cf. A238860 (partitions with superdiagonal growth), A238861 (compositions with superdiagonal growth), A000009 (partitions into distinct parts have superdiagonal growth by definition).
The version for partitions is A001477, strong A002620.
The version for permutations is A057427, strong A000295.
The opposite version is A238874, first column of A352522.
The version for fixed points is A240736, nonfixed A352520.
The strong version is A351983, column k=1 of A352524.
This is column k = 1 of A352525.
A238349 counts compositions by fixed points, first col A238351.
A352517 counts weak excedances of standard compositions.
Cf. A008930, A010054, A088218, A098825, A114088, A219282, A238352, A319005, A350839, A352488, A352489, A352514, A352515, A352516.

A179748 := proc(n,k) option remember; if k= 1 then 1; elif k> n then 0 ; else add( procname(n-i,k-1),i=1..k-1) ; end if; end proc:
A177510 := proc(n) add(A179748(n,k),k=1..n) ;end proc:
seq(A177510(n),n=1..20) ; # R. J. Mathar, Dec 14 2010

Clear[t, nn]; nn = 39; t[n_, 1] = 1; t[n_, k_] := t[n, k] = If[n >= k, Sum[t[n - i, k - 1], {i, 1, k - 1}], 0]; Table[Sum[t[n, k], {k, 1, n}], {n, 1, nn}] (* Mats Granvik, Jan 01 2015 *)
pdw[y_]:=Length[Select[Range[Length[y]],#<=y[[#]]&]]; Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],pdw[#]==1&]],{n,0,10}] (* Gus Wiseman, Mar 31 2022 *)

N=66; q='q+O('q^N); Vec( 1 + q/(1-q) * sum(n=0, N, q^n * prod(k=1, n, (1-q^k)/(1-q) ) ) ) \\ Joerg Arndt, Mar 24 2014

@CachedFunction
def T(n, k): # A179748
    if n == 0:  return int(k==0);
    if k == 1:  return int(n>=1);
    return sum( T(n-i, k-1) for i in [1..k-1] );
# to display triangle A179748 including column zero = [1,0,0,0,...]:
#for n in [0..10]: print([ T(n,k) for k in [0..n] ])
def a(n): return sum( T(n,k) for k in [0..n] )
print([a(n) for n in [0..66]])
# Joerg Arndt, Mar 24 2014

G.f.: 1 + q/(1-q) * sum(n>=0, q^n * prod(k=1..n, (1-q^k)/(1-q) ) ). [Joerg Arndt, Mar 24 2014]

New name and a(0) = 1 prepended, Joerg Arndt, Mar 24 2014

A352486 Heinz numbers of non-self-conjugate integer partitions.

Original entry on oeis.org

3, 4, 5, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73

Offset: 1

Gus Wiseman, Mar 20 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 sequence lists all Heinz numbers of partitions whose Heinz number is different from that of their conjugate.

			The terms together with their prime indices begin:
   3: (2)
   4: (1,1)
   5: (3)
   7: (4)
   8: (1,1,1)
  10: (3,1)
  11: (5)
  12: (2,1,1)
  13: (6)
  14: (4,1)
  15: (3,2)
  16: (1,1,1,1)
  17: (7)
  18: (2,2,1)
For example, the self-conjugate partition (4,3,3,1) has Heinz number 350, so 350 is not in the sequence.

The complement is A088902, counted by A000700.
These partitions are counted by A330644.
These are the positions of nonzero terms in A352491.
A000041 counts integer partitions, strict A000009.
A098825 counts permutations by unfixed points.
A238349 counts compositions by fixed points, rank statistic A352512.
A325039 counts partitions w/ same product as conjugate, ranked by A325040.
A352523 counts compositions by unfixed points, rank statistic A352513.
Heinz number (rank) and partition:
- A003963 = product of partition, conjugate A329382
- A008480 = number of permutations of partition, conjugate A321648.
- A056239 = sum of partition
- A122111 = rank of conjugate 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, A026424, A120383, A175508, A195017, A238745, A301987, A304360, A316524,  A324846, A350841.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
conj[y_]:=If[Length[y0]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Select[Range[100],#!=Times@@Prime/@conj[primeMS[#]]&]

a(n) != A122111(a(n)).

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).

A352525 Irregular triangle read by rows where T(n,k) is the number of integer compositions of n with k weak excedances (parts on or above the diagonal), all zeros removed.

Original entry on oeis.org

1, 1, 2, 3, 1, 5, 3, 8, 8, 14, 17, 1, 25, 35, 4, 46, 70, 12, 87, 137, 32, 167, 268, 76, 1, 324, 525, 170, 5, 634, 1030, 367, 17, 1248, 2026, 773, 49, 2466, 3999, 1598, 129, 4887, 7914, 3267, 315, 1, 9706, 15695, 6631, 730, 6, 19308, 31181, 13393, 1631, 23

Offset: 0

Gus Wiseman, Mar 22 2022

			Triangle begins:
     1
     1
     2
     3     1
     5     3
     8     8
    14    17     1
    25    35     4
    46    70    12
    87   137    32
   167   268    76     1
   324   525   170     5
For example, row n = 6 counts the following compositions:
  (6)       (15)     (123)
  (51)      (24)
  (312)     (33)
  (411)     (42)
  (1113)    (114)
  (1122)    (132)
  (2112)    (141)
  (2121)    (213)
  (3111)    (222)
  (11112)   (231)
  (11121)   (321)
  (11211)   (1131)
  (21111)   (1212)
  (111111)  (1221)
            (1311)
            (2211)
            (12111)

Andrew Howroyd, Table of n, a(n) for n = 0..2489 (rows 0..200)
MathOverflow, Why 'excedances' of permutations? [closed].

Row sums are A011782.
The version for partitions is A115994.
The version for permutations is A123125, strong A173018.
Column k = 1 is A177510.
The corresponding rank statistic is A352517.
The strong opposite is A352521, first col A219282, rank statistic A352514.
The opposite version is A352522, first col A238874, rank statistic A352515.
The strong version is A352524, first column A008930, rank statistic A352516.
A008292 is the triangle of Eulerian numbers (version without zeros).
A238349 counts comps by fixed points, first col A238351, rank stat A352512.
A352489 lists the weak excedance set of A122111.
A352523 counts comps by unfixed points, first A352520, rank stat A352513.
Cf. A088218, A098825, A114088, A238352, A319005, A350839, A352488.

pdw[y_]:=Length[Select[Range[Length[y]],#<=y[[#]]&]];
DeleteCases[Table[Length[Select[Join@@ Permutations/@IntegerPartitions[n],pdw[#]==k&]],{n,0,10},{k,0,n}],0,{2}]

T(n)={my(v=vector(n+1, i, i==1), r=v); for(k=1, n, v=vector(#v, j, sum(i=1, j-1, if(k<=i,x,1)*v[j-i])); r+=v); r[1]=x; [Vecrev(p) | p<-r/x]}
{ my(A=T(10)); for(i=1, #A, print(A[i])) } \\ Andrew Howroyd, Jan 19 2023

cp's OEIS Frontend

A008290 Triangle T(n,k) of rencontres numbers (number of permutations of n elements with k fixed points).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Formula

Extensions

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

A238349 Triangle T(n,k) read by rows: T(n,k) is the number of compositions of n with k parts p at position p (fixed points), n>=0, 0<=k<=n.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A177510 Number of compositions (p0, p1, p2, ...) of n with pi - p0 <= i and pi >= p0.

Views

Author