A377823 -id:A377823 - cp's OEIS frontend

A377824 Sum of the positions of minimum parts in all compositions of n.

0, 1, 4, 10, 29, 70, 181, 435, 1046, 2470, 5762, 13283, 30371, 68847, 154935, 346433, 770154, 1703152, 3748574, 8214805, 17931172, 38997819, 84531066, 182661514, 393578129, 845777569, 1813017039, 3877390908, 8274351482, 17621535902, 37456091552, 79472869966

Offset: 0

John Tyler Rascoe, Nov 08 2024

			The composition of 7, (1,2,1,1,2) has minimum parts at positions 1, 3, and 4; so it contributes 8 to a(7) = 435.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A001792, A010054, A011782, A097976, A097979, A377823.

b:= proc(n, i, p) option remember; `if`(i<1, 0,
      `if`(irem(n, i)=0, (j-> (p+j)!/j!*(p+j+1)/2*j)(n/i), 0)+
      add(b(n-i*j, i-1, p+j)/j!, j=0..(n-1)/i))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..31);  # Alois P. Heinz, Nov 12 2024

b[n_, i_, p_] := b[n, i, p] = If[i < 1, 0, If[Mod[n, i] == 0, Function[j, (p + j)!/j!*(p + j + 1)/2*j][n/i], 0] + Sum[b[n - i*j, i - 1, p + j]/j!, {j, 0, (n - 1)/i}]];
a[n_] := b[n, n, 0];
Table[a[n], {n, 0, 31}] (* Jean-François Alcover, Apr 19 2025, after Alois P. Heinz *)

A_xy(N) = {my(x='x+O('x^N), h = sum(m=1,N, sum(i=1,N, ((y^i)*x^m)*((x^(m+1))/(1-x))^(i-1)*(sum(j=0,N-m-i, prod(u=1,j, (x^(m+1))/(1-x)+(y^(u+i))*x^m)))))); h}
P_xy(N) = Pol(A_xy(N), {x})
A_x(N) = {my(px = deriv(P_xy(N),y), y=1); Vecrev(eval(px))}
A_x(20)

G.f.: A(x) = d/dy A(x,y)|{y = 1}, where A(x,y) = Sum{m>0} (Sum_{i>0} (x^m * y^i * (x^(m+1)/(1-x))^(i-1) * (Sum_{j>=0} (Product_{u=1..j} (x^(m+1)/(1-x) + x^m * y^(u+i)) ) ) ) ).

Conjecture: a(n) ~ n^2 * 2^(n-5). - Vaclav Kotesovec, Apr 19 2025

A379836 Number of pairs of adjacent equal parts in all complete compositions of n.

Original entry on oeis.org

0, 0, 1, 2, 5, 12, 23, 54, 118, 258, 550, 1178, 2540, 5394, 11473, 24174, 51021, 107210, 225099, 471322, 985202, 2055542, 4281847, 8906676, 18500425, 38379246, 79516158, 164561560, 340179441, 702506576, 1449311429, 2987297778, 6151964642, 12658841766, 26027603925

Offset: 0

John Tyler Rascoe, Jan 14 2025

nonn

An integer composition is complete if its set of parts covers an initial interval.

			The complete compositions of n = 4 are: (1,1,2), (1,2,1), (2,1,1), and (1,1,1,1); having a total of 5 pairs of equal adjacent parts giving a(4) = 5.

Cf. A011782, A106356, A107428, A107429, A373306, A374147, A374726, A377823.

C_xz(s,N) = {my(x='x+O('x^N), g=if(#s <1,1, sum(i=1,#s, C_xz(s[^i],N+1) * x^(s[i])/(1-(x^(s[i]))*(z-1)) )/(1-sum(i=1,#s, x^(s[i])/(1-(x^(s[i]))*(z-1)))))); return(g)}
B_xz(N) = {my(x='x+O('x^N), j=1, h=0); while((j*(j+1))/2 <= N, h += C_xz(vector(j, i, i), N+1); j+=1); h}
P_xz(N) = Pol(B_xz(N), {x})
B_x(N) = {my(cx = deriv(P_xz(N),z), z=1); Vecrev(eval(cx))}
B_x(20)

G.f.: B(x) = d/dz Sum_{k>0} C({1..k},x,z)|{z=1} where C({s},x,z) = Sum{i in {s}} ( C({s}-{i},x,z)*(x^i)/(1-(x^i)*(z-1)) )/(1 - Sum_{i in {s}} (x^i)/(1-(x^i)*(z-1))) with C({},x,z) = 1.

A380176 Number of pairs of adjacent equal parts in all gap-free compositions of n.

Original entry on oeis.org

0, 0, 1, 2, 6, 12, 26, 56, 124, 266, 563, 1204, 2573, 5468, 11559, 24370, 51281, 107720, 225867, 472660, 987378, 2059180, 4287932, 8916624, 18517398, 38406486, 79563118, 164636582, 340308519, 702713844, 1449664783, 2987870476, 6152930738, 12660419370, 26030245642

Offset: 0

John Tyler Rascoe, Jan 14 2025

nonn

An integer composition is gap-free if its set of parts covers an interval.

			The gap-free compositions of n = 4 are: (4), (2,2), (1,1,2), (1,2,1), (2,1,1), and (1,1,1,1); having a total of 6 pairs of equal adjacent parts giving a(4) = 6.

Cf. A011782, A106356, A107428, A107429, A373306, A374147, A374726, A377823.

C_xz(s,N) = {my(x='x+O('x^N), g=if(#s <1,1, sum(i=1,#s, C_xz(s[^i],N+1) * x^(s[i])/(1-(x^(s[i]))*(z-1)) )/(1-sum(i=1,#s, x^(s[i])/(1-(x^(s[i]))*(z-1)))))); return(g)}
B_xz(N) = {my(x='x+O('x^N), j=1, h=0); while((j*(j+1))/2 <= N, for(k=0,N, h += C_xz([(1+k)..(j+k)], N+1)); j+=1); h}
P_xz(N) = Pol(B_xz(N), {x})
B_x(N) = {my(cx = deriv(P_xz(N),z), z=1); Vecrev(eval(cx))}
B_x(20)

G.f.: B(x) = d/dz Sum_{j>0} Sum_{k>=j} C({j..k},x,z)|{z=1} where C({s},x,z) = Sum{i in {s}} ( C({s}-{i},x,z)*(x^i)/(1-(x^i)*(z-1)) )/(1 - Sum_{i in {s}} (x^i)/(1-(x^i)*(z-1))) with C({},x,z) = 1.

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A377824 Sum of the positions of minimum parts in all compositions of n.

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A379836 Number of pairs of adjacent equal parts in all complete compositions of n.

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A380176 Number of pairs of adjacent equal parts in all gap-free compositions of n.

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