A075834 -id:A075834 - cp's OEIS frontend

A363432 Number of 231-avoiding stabilized-interval-free permutations of size n.

1, 1, 1, 1, 2, 6, 18, 54, 170, 551, 1817, 6092, 20722, 71325, 248055, 870402, 3077861, 10959008, 39261382, 141430953, 512002865, 1861872379, 6798330676, 24915934639, 91630864177, 338048560865, 1250793108398, 4640542045919, 17260221009367, 64349394615738, 240434325753052

Offset: 0

Juan B. Gil, Jun 22 2023

nonn

A stabilized-interval-free (SIF) permutation on [n] = {1, 2, ..., n} is one that does not stabilize any proper subinterval of [n].

a(n) is also the number of 312-avoiding SIF permutations of size n.

			For n=5 the a(5)=6 permutations are 51234, 51423, 53124, 54123, 54132, 54213.

Daniel Birmajer, Juan B. Gil, Jordan O. Tirrell, and Michael D. Weiner, Pattern-avoiding stabilized-interval-free permutations, arXiv:2306.03155 [math.CO], 2023.

Cf. A000108, A075834.

nmax = 30; CoefficientList[Series[1 + x/(1 + CatalanNumber[1]*x^2*(x + 1) + ContinuedFractionK[-x, 1 + CatalanNumber[k]*x^(k + 1)*(x + 1), {k, 2, nmax}]), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 23 2023 *)

G.f.: 1 + x/(1+C(1)*x^2*(x+1)-x/(1+C(2)*x^3*(x+1)-x/(1+C(3)*x^4*(x+1)-x/(...)))), where C(k)=binomial(2*k,k)/(k+1).

A363433 Number of (123,231)-avoiding stabilized-interval-free permutations of size n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 3, 5, 5, 7, 7, 10, 9, 13, 12, 16, 15, 20, 18, 24, 22, 28, 26, 33, 30, 38, 35, 43, 40, 49, 45, 55, 51, 61, 57, 68, 63, 75, 70, 82, 77, 90, 84, 98, 92, 106, 100, 115, 108, 124, 117, 133, 126, 143, 135, 153, 145, 163, 155, 174, 165, 185, 176, 196

Offset: 0

Juan B. Gil, Jun 30 2023

A stabilized-interval-free (SIF) permutation on [n] = {1, 2, ..., n} is one that does not stabilize any proper subinterval of [n].

			For n from 1 to 5 the six permutations (1+1+1+1+2) are 1, 21, 312, 4312, 54132, 54213.

Paolo Xausa, Table of n, a(n) for n = 0..10000
Daniel Birmajer, Juan B. Gil, Jordan O. Tirrell, and Michael D. Weiner, Pattern-avoiding stabilized-interval-free permutations, arXiv:2306.03155 [math.CO], 2023.
Index entries for linear recurrences with constant coefficients, signature (0,2,1,-1,-2,0,1).

Cf. A075834, A363431, A363432.

A131713 := proc(n)
    op(1+modp(n,3),[1,-2,1]) ;
end proc:
A363433 := proc(n)
    if n < 3 then
        1;
    else
        16*A131713(n) +42*n-79+6*n^2-81*(-1)^n+18*n*(-1)^n;
        %/144 ;
    end if;
end proc:
seq(A363433(n),n=0..20) ; # R. J. Mathar, Jul 17 2023

LinearRecurrence[{0,2,1,-1,-2,0,1},{1,1,1,1,1,2,3,3,5,5},100] (* Paolo Xausa, Nov 18 2023 *)

Vec((x^9 + x^8 - 3*x^6 - 2*x^5 + x^4 + 2*x^3 + x^2 - x - 1)/((x^2 + x + 1)*(x + 1)^2*(x - 1)^3) + O(x^65)) \\ Michel Marcus, Jul 01 2023

G.f.: (x^9 + x^8 - 3*x^6 - 2*x^5 + x^4 + 2*x^3 + x^2 - x - 1)/((x^2 + x + 1)*(x + 1)^2*(x - 1)^3).
E.g.f.: (144 + 36*x*(2 + x) + (3*x^2 + 15*x - 80)*cosh(x) + 8*exp(-x/2)*(cos(sqrt(3)*x/2) - sqrt(3)*sin(sqrt(3)*x/2)) + (3*x^2 + 33*x + 1)*sinh(x))/72. - Stefano Spezia, Jul 01 2023
144*a(n) = 16*A131713(n) +42*n -79 +6*n^2 -81*(-1)^n +18*n*(-1)^n , for n>=3. - R. J. Mathar, Jul 17 2023

A385287 a(0) = 1, a(1) = 0; a(n) = a(n-2) + Sum_{k=0..n-1} k * a(k) * a(n-1-k).

Original entry on oeis.org

1, 0, 1, 2, 7, 32, 177, 1148, 8535, 71552, 668037, 6877742, 77448741, 947342072, 12512378625, 177525399952, 2693306735145, 43516930747192, 746123462304725, 13531269497675506, 258807528403312427, 5206929233591435496, 109929366336996502793, 2430108139669253103756

Offset: 0

Seiichi Manyama, Jul 22 2025

nonn

Cf. A075834, A386467.

Cf. A082582.

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=2, n, v[i+1]=v[i-1]+sum(j=0, i-1, j*v[j+1]*v[i-j])); v;

G.f. A(x) satisfies A(x) = 1/( 1 - x^2 - x^2 * (d/dx A(x)) ).

A386467 a(0) = a(1) = 1; a(n) = a(n-1) + a(n-2) + Sum_{k=0..n-1} k * a(k) * a(n-1-k).

Original entry on oeis.org

1, 1, 3, 11, 56, 353, 2619, 22175, 210077, 2196732, 25104008, 311139385, 4156661566, 59551385285, 910955221547, 14821776943015, 255639834413712, 4659720389150655, 89515541970546889, 1807824383345511646, 38294715773270374886, 849051935815301595992, 19665430140710069083996

Offset: 0

Seiichi Manyama, Jul 22 2025

nonn

Cf. A075834, A385287.

Cf. A307733.

a_vector(n) = my(v=vector(n+1, i, if(i<=2, 1, 0))); for(i=2, n, v[i+1]=v[i]+v[i-1]+sum(j=0, i-1, j*v[j+1]*v[i-j])); v;

G.f. A(x) satisfies A(x) = 1/( 1 - x - x^2 - x^2 * (d/dx A(x)) ).

A091139 Row sums of triangle A091063.

Original entry on oeis.org

1, 1, 2, 5, 16, 66, 346, 2229, 17000, 148898, 1465364, 15957314, 190158712, 2459041744, 34278016954, 512253587397, 8168812190472, 138450960309882

Offset: 0

Paul D. Hanna, Dec 21 2003

nonn

The initial terms of the binomial transform of the n-th row of triangle A091063 forms the n-th row of triangle A059438 transposed (permutations of [1..n] with k components); the row sums of A059438 equal the factorials.

Cf. A091063.

G.f.: A(x) = 1/(1-x*g(x)) where g(x) is the g.f. of A075834 shift 1 place left.

A134988 Number of formal expressions obtained by applying iterated binary brackets to n indexed symbols x_1, ..., x_n such that: 1) each symbol appears exactly once; 2) the smallest index inside a bracket appears on the left hand side and the largest index appears on the right hand side; 3) the outer bracket is the only bracket whose set of indices is a sequence of consecutive integers.

Original entry on oeis.org

1, 0, 1, 4, 22, 144, 1089, 9308, 88562, 927584, 10603178, 131368648, 1753970380, 25112732512, 383925637137, 6243618722124, 107644162715098, 1961478594977856, 37671587406585006, 760654555198989240, 16110333600696417780, 357148428086308848480, 8271374327887650503130

Offset: 2

Paolo Salvatore and Roberto Tauraso, Feb 05 2008, Feb 22 2008

nonn

a(n) is the number of generators in arity n of the operad Lie, when considered as a free non-symmetric operad.

Francis Brown and Jonas Bergström, Inversion of series and the cohomology of the moduli spaces m_(0,n)^δ, arXiv:0910.0120 [math.AG], 2009.
Jesse Elliott, Asymptotic expansions of the prime counting function, arXiv:1809.06633 [math.NT], 2018.
P. Salvatore and R. Tauraso, The Operad Lie is Free, arXiv:0802.3010 [math.QA], 2008.

Cf. A075834.

terms = 23; F[x_] = Sum[n! x^n, {n, 0, terms+1}]; CoefficientList[(x - InverseSeries[Series[x F[x], {x, 0, terms+1}], x])/x^2, x] (* Jean-François Alcover, Feb 17 2019 *)

N=66;  x='x+O('x^N);
F = sum(n=0,N,x^n*n!);
gf= x - serreverse(x*F);  Vec(Ser(gf))
/* Joerg Arndt, Mar 07 2013 */

a(2) = 1, a(n) = Sum_{k=2..n-2} ((k+1)*a(k+1) + a(k))*a(n-k), n > 2;
G.f.: x - series_reversion(x*F(x)), where F(x) is the g.f. of the factorials (A000142).
a(n) = (1/e)*(1 - 3/n - 5/(2n^2) + O(1/n^3)).

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A363432 Number of 231-avoiding stabilized-interval-free permutations of size n.

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A363433 Number of (123,231)-avoiding stabilized-interval-free permutations of size n.

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A385287 a(0) = 1, a(1) = 0; a(n) = a(n-2) + Sum_{k=0..n-1} k * a(k) * a(n-1-k).

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A386467 a(0) = a(1) = 1; a(n) = a(n-1) + a(n-2) + Sum_{k=0..n-1} k * a(k) * a(n-1-k).

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A091139 Row sums of triangle A091063.

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