A123125 -id:A123125 - cp's OEIS frontend

A352524 Irregular triangle read by rows where T(n,k) is the number of integer compositions of n with k excedances (parts above the diagonal), all zeros removed.

1, 1, 1, 1, 2, 2, 3, 5, 6, 9, 1, 11, 18, 3, 21, 35, 8, 41, 67, 20, 80, 131, 44, 1, 157, 257, 94, 4, 310, 505, 197, 12, 614, 996, 406, 32, 1218, 1973, 825, 80, 2421, 3915, 1669, 186, 1, 4819, 7781, 3364, 415, 5, 9602, 15486, 6762, 901, 17, 19147, 30855, 13567, 1918, 49

Offset: 0

Gus Wiseman, Mar 22 2022

			Triangle begins:
     1
     1
     1     1
     2     2
     3     5
     6     9     1
    11    18     3
    21    35     8
    41    67    20
    80   131    44     1
   157   257    94     4
   310   505   197    12
   614   996   406    32
For example, row n = 5 counts the following compositions:
  (113)    (5)     (23)
  (122)    (14)
  (1112)   (32)
  (1121)   (41)
  (1211)   (131)
  (11111)  (212)
           (221)
           (311)
           (2111)

Andrew Howroyd, Table of n, a(n) for n = 0..2507 (rows 0..200)
MathOverflow, Why 'excedances' of permutations? [closed].

The version for permutations is A008292, weak A123125.
Column k = 0 is A008930.
Row sums are A011782.
The opposite version for partitions is A114088.
The weak version for partitions is A115994.
Column k = 1 is A351983.
The corresponding rank statistic is A352516.
The opposite version is A352521, first col A219282, rank statistic A352514.
The weak opposite version is A352522, first col A238874, rank stat A352515.
The weak version is A352525, first col (k = 1) A177510, rank stat A352517.
A238349 counts comps by fixed points, first col A238351, rank stat A352512.
A352487 lists the excedance set of A122111, opposite A352490.
A352523 counts comps by unfixed points, first A352520, rank stat A352513.
Cf. A088218, A098825, A238352, A350839.

pd[y_]:=Length[Select[Range[Length[y]],#

S(v,u)={vector(#v, k, sum(i=1, k-1, v[k-i]*u[i]))}
T(n)={my(v=vector(1+n), s); v[1]=1; s=v; for(i=1, n, v=S(v, vector(n, j, if(j>i,'x,1))); s+=v); [Vecrev(p) | p<-s]}
{ my(A=T(12)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Jan 02 2023

A008518 Triangle of Eulerian numbers with rows multiplied by 1 + x.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 5, 5, 1, 1, 12, 22, 12, 1, 1, 27, 92, 92, 27, 1, 1, 58, 359, 604, 359, 58, 1, 1, 121, 1311, 3607, 3607, 1311, 121, 1, 1, 248, 4540, 19912, 31238, 19912, 4540, 248, 1, 1, 503, 15110, 102842, 244424, 244424, 102842, 15110, 503, 1

Offset: 0

N. J. A. Sloane

			Triangle begins:
   1;
   1,   1;
   1,   2,    1;
   1,   5,    5,    1;
   1,  12,   22,   12,    1;
   1,  27,   92,   92,   27,    1;
   1,  58,  359,  604,  359,   58,   1;
   1, 121, 1311, 3607, 3607, 1311, 121, 1;
   ...

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 243.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 254.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 215.

Alois P. Heinz, Rows n = 0..140, flattened
Huyile Liang, Yanni Pei, and Yi Wang, Analytic combinatorics of coordination numbers of cubic lattices, arXiv:2302.11856 [math.CO], 2023. See p. 22.

Cf. A000007, A008292, A098558 (row sums), A177042 (T(2*n, n)).

Columns include A000325 (k=1).

Eulerian:= func< n, k | (&+[(-1)^j*Binomial(n+1, j)*(k-j+1)^n: j in [0..k+1]]) >;
[Eulerian(n, k-1) + Eulerian(n, k): k in [0..n], n in [0..10]]; // G. C. Greubel, Jun 18 2024

t[n_ /; n >= 0, 0] = 1; t[n_, k_] /; k<0 || k>n = 0; t[n_, k_] := t[n, k] = (n-k) t[n-1, k-1] + (k+1) t[n-1, k];
A[n_, k_] /; k == n+1 = 0; A[n_, k_] := t[n, n-k];
T[n_, k_] := A[n, k] + A[n, k+1];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 26 2019, after Franck Maminirina Ramaharo *)

def Eulerian(n,k): return sum((-1)^j*binomial(n+1, j)*(k-j+1)^n for j in range(k+2))
flatten([[Eulerian(n,k-1) + Eulerian(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jun 18 2024

E.g.f.: (exp(x) - y*exp(y*x))/(exp(y*x) - y*exp(x)). - Vladeta Jovovic, Apr 06 2001
T(n,k) = A123125(n,k) + A123125(n,k+1), with A123125(n,n+1) = 0. - Franck Maminirina Ramaharo, Oct 21 2018
From G. C. Greubel, Jun 18 2024: (Start)
T(n, n-k) = T(n, k).
Sum_{k=0..n} (-1)^k*T(n, k) = A000007(n). (End)

More terms from Vladeta Jovovic, Apr 06 2001

A046739 Triangle read by rows, related to number of permutations of [n] with 0 successions and k rises.

Original entry on oeis.org

0, 1, 1, 1, 1, 7, 1, 1, 21, 21, 1, 1, 51, 161, 51, 1, 1, 113, 813, 813, 113, 1, 1, 239, 3361, 7631, 3361, 239, 1, 1, 493, 12421, 53833, 53833, 12421, 493, 1, 1, 1003, 42865, 320107, 607009, 320107, 42865, 1003, 1, 1, 2025, 141549, 1704693, 5494017

Offset: 1

N. J. A. Sloane

From Emeric Deutsch, May 25 2009: (Start)
T(n,k) is the number of derangements of [n] having k excedances. Example: T(4,2)=7 because we have 3*14*2, 3*4*12, 4*3*12, 2*14*3, 2*4*13, 3*4*21, 4*3*21, each with two excedances (marked). An excedance of a permutation p is a position i such that p(i) > i.
Sum_{k>=1} k*T(n,k) = A000274(n+1). (End)
The triangle 1;1,1;1,7,1;... has general term T(n,k) = Sum_{j=0..n+2} (-1)^(n-j)*C(n+2,j)*A123125(j,k+2) and bivariate g.f. ((1-y)*(y*exp(2*x*y) + exp(x*(y+1))(y^2 - 4*y + 1) + y*exp(2*x)))/(exp(x*y) - y*exp(x))^3. - Paul Barry, May 10 2011
The n-th row is the local h-vector of the barycentric subdivision of a simplex, i.e., the Coxeter complex of type A. See Proposition 2.4 of Stanley's paper below. - Kyle Petersen, Aug 20 2012
T(n,k) is the k-th coefficient of the local h^*-polynomial, or box polynomial, of the s-lecture hall n-simplex with s=(2,3,...,n+1).  See Theorem 4.1 of the paper by N. Gustafsson and L. Solus below. - Liam Solus, Aug 23 2018

			Triangle starts:
  0;
  1;
  1,   1;
  1,   7,   1;
  1,  21,  21,   1;
  1,  51, 161,  51,   1;
  1, 113, 813, 813, 113, 1;
  ...
From _Peter Luschny_, Sep 17 2021: (Start)
The triangle shows the coefficients of the following bivariate polynomials:
  [1] 0;
  [2] x*y;
  [3] x^2*y +     x*y^2;
  [4] x^3*y +   7*x^2*y^2 +     x*y^3;
  [5] x^4*y +  21*x^3*y^2 +  21*x^2*y^3 +     x*y^4;
  [6] x^5*y +  51*x^4*y^2 + 161*x^3*y^3 +  51*x^2*y^4 +     x*y^5;
  [7] x^6*y + 113*x^5*y^2 + 813*x^4*y^3 + 813*x^3*y^4 + 113*x^2*y^5 + x*y^6;
  ...
These polynomials are the permanents of the n X n matrices with all entries above the main antidiagonal set to 'x' and all entries below the main antidiagonal set to 'y'. The main antidiagonals consist only of zeros. Substituting x <- 1 and y <- -1 generates the Euler secant numbers A122045. (Compare with A081658.)
(End)

Robert Israel, Table of n, a(n) for n = 1..10012 (rows 0 to 142, flattened)
L. Carlitz, Richard Scoville, and Theresa Vaughan, Enumeration of pairs of permutations and sequences, Bulletin of the American Mathematical Society 80.5 (1974): 881-884. [Annotated scanned copy]
L. Carlitz, N. J. A. Sloane, and C. L. Mallows, Correspondence, 1975
N. Gustafsson and L. Solus. Derangements, Ehrhart theory, and local h-polynomials, arXiv:1807.05246 [math.CO], 2018.
R. Mantaci and F. Rakotondrajao, Exceedingly deranging!, Advances in Appl. Math., 30 (2003), 177-188. [_Emeric Deutsch_, May 25 2009]
Lili Mu and Volkmar Welker, On a question about real rooted polynomials and f-polynomials of simplicial complexes, arXiv:2503.24076 [math.CO], 2025. See p. 8.
D. P. Roselle, Permutations by number of rises and successions, Proc. Amer. Math. Soc., 19 (1968), 8-16. [Annotated scanned copy]
D. P. Roselle, Permutations by number of rises and successions, Proc. Amer. Math. Soc., 20 (1968), 8-16.
R. P. Stanley, Subdivisions and local h-vectors, J. Amer. Math. Soc., 5 (1992), 805-851.
John D. Wiltshire-Gordon, Alexander Woo, and Magdalena Zajaczkowska, Specht Polytopes and Specht Matroids, arXiv:1701.05277 [math.CO], 2017. [See Conjecture 6.2]

Cf. A046740.
Row sums give A000166.
Diagonals give A070313, A070315.
T(2n,n) gives A320337.
Cf. A000274, A081658, A122045, A271697.

G := (1-t)*exp(-t*z)/(1-t*exp((1-t)*z)): Gser := simplify(series(G, z = 0, 15)): for n to 13 do P[n] := sort(expand(factorial(n)*coeff(Gser, z, n))) end do: 0; for n to 11 do seq(coeff(P[n], t, j), j = 1 .. n-1) end do; # yields sequence in triangular form # Emeric Deutsch, May 25 2009

max = 12; f[t_, z_] := (1-t)*(Exp[-t*z]/(1 - t*Exp[(1-t)*z])); se = Series[f[t, z], {t, 0, max}, {z, 0, max}];
coes = Transpose[ #*Range[0, max]! & /@ CoefficientList[se, {t, z}]]; Join[{0}, Flatten[ Table[ coes[[n, k]], {n, 2, max}, {k, 2, n-1}]]] (* Jean-François Alcover, Oct 24 2011, after g.f. *)
E1[n_ /; n >= 0, 0] = 1; (* E1(n,k) are the Eulerian numbers *)
E1[n_, k_] /; k < 0 || k > n = 0;
E1[n_, k_] := E1[n, k] = (n-k) E1[n-1, k-1] + (k+1) E1[n-1, k];
T[n_, k_] := Sum[Binomial[-j-1, -n-1] E1[j, k], {j, 0, n}];
Table[T[n, k], {n, 1, 100}, {k, 1, n-1}] /. {} -> {0} // Flatten (* Jean-François Alcover, Oct 31 2020, after Peter Luschny in A271697 *)
Table[Expand[n!Factor[SeriesCoefficient[(x-y)/(x Exp[y t]-y Exp[x t]),{t,0,n}]]],{n,0,12}]//TableForm (* Mamuka Jibladze, Nov 26 2024 *)

T(n)={my(x='x+O('x^(n+1))); concat([[0]], [Vecrev(p/y) | p<-Vec(-1+serlaplace((y-1)/(y*exp(x)-exp(x*y))))])}
{ my(A=T(10));for(i=1,#A,print(A[i])) } \\ Andrew Howroyd, Nov 13 2024

a(n+1, r) = r*a(n, r) + (n+1-r)*a(n, r-1) + n*a(n-1, r-1).
exp(-t)/(1 - exp((x-1)t)/(x-1)) = 1 + x*t^2/2! + (x+x^2)*t^3/3! + (x+7x^2+x^3)*t^4/4! + (x+21x^2+21x^3+x^4)*t^5/5! + ... - Philippe Deléham, Jun 11 2004
E.g.f.: (y-1)/(y*exp(x) - exp(x*y)). - Mamuka Jibladze, Nov 08 2024

More terms from Larry Reeves (larryr(AT)acm.org), Apr 07 2000

A340556 E2(n, k), the second-order Eulerian numbers with E2(0, k) = δ_{0, k}. Triangle read by rows, E2(n, k) for 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 1, 8, 6, 0, 1, 22, 58, 24, 0, 1, 52, 328, 444, 120, 0, 1, 114, 1452, 4400, 3708, 720, 0, 1, 240, 5610, 32120, 58140, 33984, 5040, 0, 1, 494, 19950, 195800, 644020, 785304, 341136, 40320, 0, 1, 1004, 67260, 1062500, 5765500, 12440064, 11026296, 3733920, 362880

Offset: 0

Peter Luschny, Feb 05 2021

The second-order Eulerian number E2(n, k) is the number of Stirling permutations of order n with exactly k descents; here the last index is defined to be a descent. More formally, let Q_n denote the set of permutations of the multiset {1,1,2,2, ..., n,n} in which, for all j, all entries between two occurrences of j are larger than j, then E2(n, k) = card({s in Q_n with des(s) = k}), where des(s) = card({j: s(j) > s(j+1)}) is the number of descents of s.
Also the number of Riordan trapezoidal words of length n with k distinct letters (see Riordan 1976, p. 9).
Also the number of rooted plane trees on n + 1 vertices with k leaves (see Janson 2008, p. 543).
Let b(n) = (1/2)*Sum_{k=0..n-1} (-1)^k*E2(n-1, k+1) / C(2*n-1, k+1). Apparently b(n) = Bernoulli(n, 1) = -n*Zeta(1 - n) = Integral_{x=0..1}F_n(x) for n >= 1. Here F_n(x) are the signed Fubini polynomials (A278075). (See Rzadkowski and Urlinska, example 4.)

			Triangle starts:
  [0] 1;
  [1] 0, 1;
  [2] 0, 1, 2;
  [3] 0, 1, 8,    6;
  [4] 0, 1, 22,   58,    24;
  [5] 0, 1, 52,   328,   444,     120;
  [6] 0, 1, 114,  1452,  4400,    3708,    720;
  [7] 0, 1, 240,  5610,  32120,   58140,   33984,    5040;
  [8] 0, 1, 494,  19950, 195800,  644020,  785304,   341136,   40320;
  [9] 0, 1, 1004, 67260, 1062500, 5765500, 12440064, 11026296, 3733920, 362880.
To illustrate the generating function for row 3: The expansion of (1 - x)^7*(x*exp(-x) + 16*x^2*exp(-x)^2 + (243*x^3*exp(-x)^3)/2) gives the polynomial x + 8*x^2 + 6*x^3. The coefficients of this polynomial give row 3.
.
Stirling permutations of order 3 with exactly k descents: (When counting the descents one may assume an invisible '0' appended to the permutations.)
  T[3, k=0]:
  T[3, k=1]: 112233;
  T[3, k=2]: 331122; 223311; 221133; 133122; 122331; 122133; 113322; 112332;
  T[3, k=3]: 332211; 331221; 233211; 221331; 133221; 123321.

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, 2nd ed. Addison-Wesley, Reading, MA, 1994, p. 270.

J. Fernando Barbero G., Jesús Salas, and Eduardo J. S. Villaseñor, Generalized Stirling permutations and forests: Higher-order Eulerian and Ward numbers, Electronic Journal of Combinatorics 22(3), 2015.
Kenny Barrese, Jennifer Elder, Pamela E. Harris, and Anthony Simpson, Enumerating Flat Fubini Rankings, arXiv:2504.06466 [math.CO], 2025. Conjecture 4.4. p. 14.
Brian Drake, An inversion theorem for labeled trees and some limits of areas under lattice paths (Example 1.10.1), Thesis, Brandeis Univ., Aug. 2008, p. 58.
Sergi Elizalde, Descents on quasi-Stirling permutations, arXiv:2002.00985 [math.CO], 2020.
I. Gessel and R. P. Stanley, Stirling polynomials, J. Combin. Theory, A 24, 24-33, 1978.
Svante Janson, Plane recursive trees, Stirling permutations and an urn model, Fifth Colloquium on Mathematics and Computer Science, 541-547, Discrete Math. Theor. Comput. Sci. Proc., AI, 2008.
Peter Luschny, A companion to A340556, A SageMath-Jupyter notebook, Feb. 2021.
Peter Luschny, How are the Eulerian numbers of the first-order related to the Eulerian numbers of the second-order?, MathOverflow, Feb. 2021.
Andrew Elvey Price and Alan D. Sokal, Phylogenetic trees, augmented perfect matchings, and a Thron-type continued fraction (T-fraction) for the Ward polynomials, arXiv:2001.01468 [math.CO], 2020.
John Riordan, The blossoming of Schröder's fourth problem, Acta Math., 137 (1976), 1-16.
G. Rzadkowski and M. Urlinska, A Generalization of the Eulerian Numbers, arXiv:1612.06635 [math.CO], 2016.

Indexing the second-order Eulerian numbers comes in three flavors: A008517 (following Riordan and Comtet), A201637 (following Graham, Knuth, and Patashnik) and this indexing, extending the definition of Gessel and Stanley. (A008517 is the main entry of the numbers.) The corresponding triangles of the first-order Eulerian numbers are A008292, A173018, and A123125.
Row reversed: A163936 (with offset = 0).
Values: E2poly(n, 1) = A001147(n), E2poly(n, -1) ~ -A001662(n+1), E2poly(n, 2) = A112487(n), 2^n*E2poly(n, 1/2) = A000311(n+1), 2^n*E2poly(n, -1/2) = A341106(n).
Diagonals: A000142, A002538, A002539, A112008, A112485.
Columns: A000007, A000012, A005803, A004301, A006260.
Cf. A008299, A137375, A008306, A106828, A269939, A278075, A331998.

# Using the recurrence:
E2 := proc(n, k) option remember;
if k = 0 and n = 0 then return 1 fi; if n < 0 then return 0 fi;
E2(n-1, k)*k + E2(n-1, k-1)*(2*n - k) end: seq(seq(E2(n, k), k = 0..n), n = 0..9);
# Using the row generating function:
E2egf := n -> (1-x)^(2*n+1)*add(k^(n+k)/k!*(x*exp(-x))^k, k=0..n);
T := (n, k) -> coeftayl(E2egf(n), x=0, k): seq(print(seq(T(n, j),j=0..n)), n=0..7);
# Using the built-in function:
E2 := (n, k) -> `if`(k=0, k^n, combinat:-eulerian2(n, k-1)):
# Using the compositional inverse (series reversion):
E2triangle := proc(N) local r, s, C; Order := N + 2;
s := solve(y = series(x - t*(exp(x) - 1), x), x):
r := n -> -n!*(t - 1)^(2*n - 1)*coeff(s, y, n); C := [seq(expand(r(n)), n = 1..N)];
seq(print(seq(coeff(C[n+1], t, k), k = 0..n)), n = 0..N-1) end: E2triangle(10);

T[n_, k_] := T[n, k] = If[k == 0, Boole[n == 0], If[n < 0, 0, k T[n - 1, k] + (2 n - k) T[n - 1, k - 1]]]; Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten
(* Via row polynomials: *)
E2poly[n_] := If[n == 0, 1,
  Expand[Simplify[x (x - 1)^(2 n) D[((1 - x)^(1 - 2 n) E2poly[n - 1]), x]]]];
Table[CoefficientList[E2poly[n], x], {n, 0, 9}] // Flatten
(* Series reversion *)
Revert[gf_, len_] := Module[{S = InverseSeries[Series[gf, {x, 0, len + 1}], x]},
Table[CoefficientList[(n + 1)! (1 - t)^(2 n + 1) Coefficient[S, x, n + 1], t],
{n, 0, len}] // Flatten]; Revert[x + t - t Exp[x], 6]

E2poly(n) = if(n == 0, 1, x*(x-1)^(2*n)*deriv((1-x)^(1-2*n)*E2poly(n-1)));
{ for(n = 0, 9, print(Vecrev(E2poly(n)))) }

T(n, k) = sum(j=0, n-k, (-1)^(n-j)*binomial(2*n+1, j)*stirling(2*n-k-j+1, n-k-j+1, 1)); \\ Michel Marcus, Feb 11 2021

# See also link to notebook.
@cached_function
def E2(n, k):
    if n < 0: return 0
    if k == 0: return k^n
    return k * E2(n - 1, k) + (2*n - k) * E2(n - 1, k - 1)  # Peter Luschny, Mar 08 2025

E2(n, k) = E2(n-1, k)*k + E2(n-1, k-1)*(2*n - k) for n > 0 and 0 <= k <= n, and E2(0, 0) = 1; in all other cases E(n, k) = 0.
E2(n, k) = Sum_{j=0..n-k}(-1)^(n-j)*binomial(2*n+1, j)*Stirling1(2*n-k-j+1, n-k-j+1).
E2(n, k) = Sum_{j=0..k}(-1)^(k-j)*binomial(2*n + 1, k - j)*Stirling2(n + j, j).
Stirling1(x, x - n) = (-1)^n*Sum_{k=0..n} E2(n, k)*binomial(x + k - 1, 2*n).
Stirling2(x, x - n) = Sum_{k=0..n} E2(n, k)*binomial(x + n - k, 2*n).
E2poly(n, x) = Sum_{k=0..n} E2(n, k)*x^k, as row polynomials.
E2poly(n, x) = x*(x-1)^(2*n)*d_{x}((1-x)^(1-2*n)*E2poly(n-1)) for n>=1 and E2poly(0)=1.
E2poly(n, x) = (1 - x)^(2*n + 1)*Sum_{k=0..n}(k^(n + k)/k!)*(x*exp(-x))^k.
W(n, k) = [x^k] (1+x)^n*E2poly(n, x/(1 + x)) are the Ward numbers A269939.
E2(n, k) = [x^k] (1-x)^n*Wpoly(n, x/(1 - x)); Wpoly(n, x) = Sum_{k=0..n}W(n, k)*x^k.
W(n, k) = Sum_{j=0..k} E2(n, j)*binomial(n - j, n - k).
E2(n, k) = Sum_{j=0..k} (-1)^(k-j)*W(n, j)*binomial(n - j, k - j).
The compositional inverse with respect to x of x - t*(exp(x) - 1) (see B. Drake):
T(n, k) = [t^k](n+1)!*(1-t)^(2*n+1)*[x^(n+1)] InverseSeries(x - t*(exp(x)-1), x).
AS1(n, k) = Sum_{j=0..n-k} binomial(j, n-2*k)*E2(n-k, j+1), where AS1(n, k) are the associated Stirling numbers of the first kind (A008306, A106828).
E2(n, k) = Sum_{j=0..n-k+1} (-1)^(n-k-j+1)*AS1(n+j, j)*binomial(n-j, n-k-j+1), for n >= 1.
AS2(n, k) = Sum_{j=0..n-k} binomial(j, n-2*k)*E2(n-k, n-k-j) for n >=1, where AS2(n, k) are the associated Stirling numbers of the second kind (A008299, A137375).
E2(n, k) = Sum_{j=0..k} (-1)^(k-j)*AS2(n + j, j)*binomial(n - j, k - j).

A352491 n minus the Heinz number of the conjugate of the integer partition with Heinz number n.

Original entry on oeis.org

0, 0, -1, 1, -3, 0, -9, 3, 0, -2, -21, 2, -51, -10, -3, 9, -111, 3, -237, 0, -15, -26, -489, 10, -2, -70, 2, -12, -995, 0, -2017, 21, -39, -158, -19, 15, -4059, -346, -105, 12, -8151, -18, -16341, -36, -5, -722, -32721, 26, -32, 5, -237, -108, -65483, 19, -53

Offset: 1

Gus Wiseman, Mar 20 2022

sign

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.

Problem: What is the image? In the nonnegative case it appears to start: 0, 1, 2, 3, 5, 7, 9, ...

			The partition (4,4,1,1) has Heinz number 196 and its conjugate (4,2,2,2) has Heinz number 189, so a(196) = 196 - 189 = 7.

Positions of zeros are A088902, counted by A000700.
A similar sequence is A175508.
Positions of nonzero terms are A352486, counted by A330644.
Positions of negative terms are A352487, counted by A000701.
Positions of nonnegative terms are A352488, counted by A046682.
Positions of nonpositive terms are A352489, counted by A046682.
Positions of positive terms are A352490, counted by A000701.
A000041 counts integer partitions, strict A000009.
A003963 is product of prime indices, conjugate A329382.
A008480 counts permutations of prime indices, conjugate A321648.
A056239 adds up prime indices, row sums of A112798 and A296150.
A122111 is partition conjugation using Heinz numbers, parts A321649/A321650.
A124010 gives prime signature, sorted A118914, length A001221, sum A001222.
A173018 counts permutations by excedances, weak A123125.
A238744 is partition conjugate of prime signature, ranked by A238745.
Cf. A000720, A114324, A301987, A324850, A325037, A325038, A325040, A347450.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Table[n-Times@@Prime/@conj[primeMS[n]],{n,30}]

a(n) = n - A122111(n).

A352490 Nonexcedance set of A122111. Numbers k > A122111(k), where A122111 represents partition conjugation using Heinz numbers.

Original entry on oeis.org

4, 8, 12, 16, 18, 24, 27, 32, 36, 40, 48, 50, 54, 60, 64, 72, 80, 81, 90, 96, 100, 108, 112, 120, 128, 135, 140, 144, 150, 160, 162, 168, 180, 192, 196, 200, 216, 224, 225, 240, 243, 250, 252, 256, 270, 280, 288, 300, 315, 320, 324, 336, 352, 360, 375, 378

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). The sequence lists all Heinz numbers of partitions whose Heinz number is greater than that of their conjugate.

			The terms together with their prime indices begin:
    4: (1,1)
    8: (1,1,1)
   12: (2,1,1)
   16: (1,1,1,1)
   18: (2,2,1)
   24: (2,1,1,1)
   27: (2,2,2)
   32: (1,1,1,1,1)
   36: (2,2,1,1)
   40: (3,1,1,1)
   48: (2,1,1,1,1)
   50: (3,3,1)
   54: (2,2,2,1)
   60: (3,2,1,1)
   64: (1,1,1,1,1,1)
For example, the partition (4,4,1,1) has Heinz number 196 and its conjugate (4,2,2,2) has Heinz number 189, and 196 > 189, so 196 is in the sequence, and 189 is not.

Alois P. Heinz, Table of n, a(n) for n = 1..2000
Richard Ehrenborg and Einar Steingrímsson, The Excedance Set of a Permutation, Advances in Applied Mathematics 24, (2000), 284-299.
MathOverflow, Why 'excedances' of permutations? [closed].

These partitions are counted by A000701.
The opposite version is A352487, weak A352489.
The weak version is A352488, counted by A046682.
These are the positions of positive terms in A352491.
A000041 counts integer partitions, strict A000009.
A000700 counts self-conjugate partitions, ranked by A088902 (cf. A258116).
A003963 = product of prime indices, conjugate A329382.
A056239 adds up prime indices, row sums of A112798 and A296150.
A122111 = partition conjugation using Heinz numbers, parts A321649/A321650.
A124010 gives prime signature, sorted A118914, length A001221, sum A001222.
A173018 counts permutations by excedances, weak A123125.
A330644 counts non-self-conjugate partitions, ranked by A352486.
A352521 counts compositions by subdiagonals, rank statistic A352514.
Cf. A000720, A114088, A114324, A321648, A324850, A325037, A325038, A325040.

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

a(n) > A122111(a(n)).

A352514 Number of strong nonexcedances (parts below the diagonal) of the n-th composition in standard order.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Mar 22 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 83rd composition in standard order is (2,3,1,1), with strong nonexcedances {3,4}, so a(83) = 2.

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

Positions of first appearances are A000225.
The weak version is A352515, counted by A352522 (first column A238874).
The opposite version is A352516, counted by A352524 (first column A008930).
The weak opposite version is A352517, counted by A352525 (first A177510).
The triangle A352521 counts these compositions (first column A219282).
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 parts, first col A238351, rank stat A352512.
A352490 is the (strong) nonexcedance set of A122111.
A352523 counts comps by unfixed parts, first col A010054, rank stat A352513.
Cf. A088218, A098825, A114088, A115994, A131577, A238352.

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

A201354 Expansion of e.g.f. exp(x) / (4 - 3*exp(x)).

Original entry on oeis.org

1, 4, 28, 292, 4060, 70564, 1471708, 35810212, 995827420, 31153998244, 1082931514588, 41407678132132, 1727226633730780, 78051253062575524, 3798351192214837468, 198049421007186054052, 11014905131587945490140, 650903915009792820650404, 40726453234725158535472348

Offset: 0

Paul D. Hanna, Nov 30 2011

			E.g.f.: E(x) = 1 + 4*x + 28*x^2/2! + 292*x^3/3! + 4060*x^4/4! + 70564*x^5/5! + ...
O.g.f.: A(x) = 1 + 4*x + 28*x^2 + 292*x^3 + 4060*x^4 + 70564*x^5 + ...
where A(x) = 1 + 4*x/(1+x) + 2!*4^2*x^2/((1+x)*(1+2*x)) + 3!*4^3*x^3/((1+x)*(1+2*x)*(1+3*x)) + 4!*4^4*x^4/((1+x)*(1+2*x)*(1+3*x)*(1+4*x)) + ...

G. C. Greubel, Table of n, a(n) for n = 0..370

Cf. A000629, A032033, A050352, A123227, A136728.

R:=PowerSeriesRing(Rationals(), 20); Coefficients(R!(Laplace( 1/(4*Exp(-x) -3) ))); // G. C. Greubel, Jun 08 2020

seq(coeff(series(1/(4*exp(-x) -3), x, n+1)*n!, x, n), n = 0..20); # G. C. Greubel, Jun 08 2020

Table[Sum[(-1)^(n-k)*4^k*StirlingS2[n,k]*k!,{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jun 13 2013 *)

{a(n)=n!*polcoeff(exp(x+x*O(x^n))/(4 - 3*exp(x+x*O(x^n))), n)}

{a(n)=polcoeff(sum(m=0, n, 4^m*m!*x^m/prod(k=1, m, 1+k*x+x*O(x^n))), n)}

{Stirling2(n, k)=if(k<0||k>n, 0, sum(i=0, k, (-1)^i*binomial(k, i)/k!*(k-i)^n))}
{a(n)=sum(k=0, n, (-1)^(n-k)*4^k*Stirling2(n, k)*k!)}

[sum( (-1)^(n-j)*4^j*factorial(j)*stirling_number2(n,j) for j in (0..n)) for n in (0..20)] # G. C. Greubel, Jun 08 2020

O.g.f.: A(x) = Sum_{n>=0} n! * 4^n*x^n / Product_{k=0..n} (1+k*x).
O.g.f.: A(x) = 1/(1 - 4*x/(1-3*x/(1 - 8*x/(1-6*x/(1 - 12*x/(1-9*x/(1 - 16*x/(1-12*x/(1 - 20*x/(1-15*x/(1 - ...))))))))))), a continued fraction.
a(n) = Sum_{k=0..n} (-1)^(n-k) * 4^k * Stirling2(n,k) * k!.
a(n) = 4*A050352(n) for n>0.
a(n) = Sum_{k=0..n} A123125(n,k)*4^k*3^(n-k). - Philippe Deléham, Nov 30 2011
a(n) = log(4/3) * Integral_{x = 0..oo} (ceiling(x))^n * (4/3)^(-x) dx. - Peter Bala, Feb 06 2015
G.f.: 2/G(0), where G(k) = 1 + 1/(1 - 8*x*(k+1)/(8*x*(k+1) - 1 + 6*x*(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 30 2013
a(n) ~ n! / (3*(log(4/3))^(n+1)). - Vaclav Kotesovec, Jun 13 2013
a(n) = 1 + 3 * Sum_{k=0..n-1} binomial(n,k) * a(k). - Ilya Gutkovskiy, Jun 08 2020
From Seiichi Manyama, Nov 15 2023: (Start)
a(0) = 1; a(n) = -4*Sum_{k=1..n} (-1)^k * binomial(n,k) * a(n-k).
a(0) = 1; a(n) = 4*a(n-1) + 3*Sum_{k=1..n-1} binomial(n-1,k) * a(n-k). (End)
a(n) = (4/3)*A032033(n) - (1/3)*0^n. - Seiichi Manyama, Dec 21 2023

A352487 Excedance set of A122111. Numbers k < A122111(k), where A122111 represents partition conjugation using Heinz numbers.

Original entry on oeis.org

3, 5, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 28, 29, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 76, 77, 78, 79, 82, 83, 85, 86, 87, 88, 89, 91, 92, 93, 94

Offset: 1

Gus Wiseman, Mar 19 2022

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). The sequence lists all Heinz numbers of partitions whose Heinz number is less than that of their conjugate.

			The terms together with their prime indices begin:
   3: (2)
   5: (3)
   7: (4)
  10: (3,1)
  11: (5)
  13: (6)
  14: (4,1)
  15: (3,2)
  17: (7)
  19: (8)
  21: (4,2)
  22: (5,1)
  23: (9)
  25: (3,3)
  26: (6,1)
  28: (4,1,1)
For example, the partition (4,1,1) has Heinz number 28 and its conjugate (3,1,1,1) has Heinz number 40, and 28 < 40, so 28 is in the sequence, and 40 is not.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Richard Ehrenborg and Einar Steingrímsson, The Excedance Set of a Permutation, Advances in Applied Mathematics 24, (2000), 284-299.
MathOverflow, Why 'excedances' of permutations? [closed].

These partitions are counted by A000701.
The weak version is A352489, counted by A046682.
The opposite version is A352490, weak A352488.
These are the positions of negative terms in A352491.
A000041 counts integer partitions, strict A000009.
A000700 counts self-conjugate partitions, ranked by A088902 (cf. A258116).
A003963 = product of prime indices, conjugate A329382.
A008292 is the triangle of Eulerian numbers (version without zeros).
A008480 counts permutations of prime indices, conjugate A321648.
A056239 adds up prime indices, row sums of A112798 and A296150.
A122111 = partition conjugation using Heinz numbers, parts A321649/A321650.
A124010 gives prime signature, sorted A118914, length A001221, sum A001222.
A173018 counts permutations by excedances, weak A123125.
A238744 = partition conjugate of prime signature, ranked by A238745.
A330644 counts non-self-conjugate partitions, ranked by A352486.
A352521 counts compositions by subdiagonals, rank statistic A352514.
Cf. A000720, A114088, A120383, A175508, A290822, A304360, A316524, A319005, A325040.

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

a(n) < A122111(a(n)).

A352515 Number of weak nonexcedances (parts on or below the diagonal) of the n-th composition in standard order.

Original entry on oeis.org

0, 1, 0, 2, 0, 1, 2, 3, 0, 1, 1, 2, 1, 3, 3, 4, 0, 1, 1, 2, 0, 2, 2, 3, 1, 2, 3, 4, 3, 4, 4, 5, 0, 1, 1, 2, 0, 2, 2, 3, 0, 1, 2, 3, 2, 3, 3, 4, 1, 2, 2, 3, 3, 4, 4, 5, 2, 4, 4, 5, 4, 5, 5, 6, 0, 1, 1, 2, 0, 2, 2, 3, 0, 1, 2, 3, 2, 3, 3, 4, 0, 1, 1, 2, 2, 3, 3

Offset: 0

Gus Wiseman, Mar 23 2022

nonn

			The 89th composition in standard order is (2,1,3,1), with weak nonexcedances {2,3,4}, so a(89) = 3.

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

Positions of first appearances are A000225.
The strong version is A352514, counted by A352521 (first column A219282).
The strong opposite version is A352516, counted by A352524 (first A008930).
The opposite version is A352517, counted by A352525 (first column A177510).
Triangle A352522 counts these comps (first col A238874), partitions A115994.
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.
A352488 is the weak nonexcedance set of A122111.
A352523 counts comps by unfixed pts, first col A010054, rank stat A352513.
Cf. A088218, A098825, A114088, A131577, A238352.

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

cp's OEIS Frontend

A352524 Irregular triangle read by rows where T(n,k) is the number of integer compositions of n with k excedances (parts above the diagonal), all zeros removed.

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A008518 Triangle of Eulerian numbers with rows multiplied by 1 + x.

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A046739 Triangle read by rows, related to number of permutations of [n] with 0 successions and k rises.

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A340556 E2(n, k), the second-order Eulerian numbers with E2(0, k) = δ_{0, k}. Triangle read by rows, E2(n, k) for 0 <= k <= n.

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A352491 n minus the Heinz number of the conjugate of the integer partition with Heinz number n.

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A352490 Nonexcedance set of A122111. Numbers k > A122111(k), where A122111 represents partition conjugation using Heinz numbers.

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A352514 Number of strong nonexcedances (parts below the diagonal) of the n-th composition in standard order.

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