User:Olivier Gérard/IntPartitions

Sequences which are of the form : "sum of f(p(i)) over all integer partitions of n".

OEIS anum	OEIS values	Formula for f
A373243	0, 0, 1, 1, 2, 5, 6, 11, 18, 27	Length Union - Union Length/@ Split

Sequences with linear row sums equal to number of partitions

OEIS anum	OEIS values	Type of sum	Name
A050314	{1, 0, 1, 1, 0, 1, 0, 1, 0, 2, 2, 0, 2, 0, 1}	A000041(n)	Triangle: a(n,k) = number of partitions of n whose xor-sum is k.
A072233	{1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1}	A000041(n)	Square array T(n,k) read by antidiagonals giving number of ways to distribute n indistinguishable objects in k indistinguishable containers; containers may be left empty.
A096651	{1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1}	A000041(n)	Lower triangular matrix T, read by rows, such that the row sums of T^n form the n-dimensional partitions.
A097567	{1, 1, 0, 0, 0, 2, 1, 0, 2, 0, 3, 0, 0, 0, 2}	A000041(n)	T(n,k)= count of partitions p such that Abs( Odd(p)-Odd(p') ) = k, where p' is the transpose of p and Odd(p) counts the odd elements in p. Related to Stanley's 'f'.
A103919	{1, 0, 1, 1, 0, 1, 0, 2, 0, 1, 2, 0, 2, 0, 1}	A000041(n)	Triangle of numbers of partitions of n with total number of odd parts equal to k from {0,...,n}.
A113685	{1, 0, 1, 1, 0, 1, 0, 1, 0, 2, 2, 0, 1, 0, 2}	A000041(n)	Triangular array read by rows: T(n,k) is the number of partitions of n in which sum of odd parts is k, for k=0,1,...,n; n>=0.
A113686	{1, 1, 0, 1, 0, 1, 2, 0, 1, 0, 2, 0, 1, 0, 2}	A000041(n)	Triangular array T(n,k)=number of partitions of n in which sum of even parts is k, for k=0,1,...n; n>=0.
A116598	{1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1}	A000041(n)	Triangle read by rows: T(n,k) is the number of partitions of n having exactly k parts equal to 1 (n>=0, 0<=k<=n).
A118198	{1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 2, 0, 1}	A000041(n)	Triangle read by rows: T(n,k) is the number of partitions of n having k parts equal to the size of the Durfee square (0<=k<=n).
A137586	{1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 3, 1, 0, 0, 1}	A000041(n)	Triangle read by rows: A026794 * A054525.
A176202	{1, 0, 1, 0, 0, 2, 1, 0, 0, 2, 0, 1, 0, 0, 4}	A000041(n)	Convolution triangle, row sums = A000041. M = A000041 in each column with two interleaved zeros; Q = A000726 diagonalized with the rest zeros. A176202 = M*Q.
A209354	{1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1}	A000041(n)	Triangular array: T(n,k) = number of partitions of n for which (maximal term)-(minimal term)=k, if 0<=k<n, and T(n,n)=1.
A238353	{1, 1, 0, 2, 0, 0, 2, 1, 0, 0, 3, 1, 1, 0, 0}	A000041(n)	Triangle T(n,k) read by rows: T(n,k) is the number of partitions of n (as weakly ascending list of parts) with maximal ascent k, n >= 0, 0 <= k <= n.
A238354	{1, 1, 0, 2, 0, 0, 2, 1, 0, 0, 4, 0, 1, 0, 0}	A000041(n)	Triangle T(n,k) read by rows: T(n,k) is the number of partitions of n (as weakly ascending list of parts) with minimal ascent k, n >= 0, 0 <= k <= n.
A243978	{1, 0, 1, 0, 1, 1, 0, 2, 0, 1, 0, 3, 1, 0, 1}	A000041(n)	Triangle T(n,k), n>=0, 0<=k<=n, read by rows: T(n,k) is the number of partitions of n where the minimal multiplicity of any part is k.
A263233	{1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1}	A000041(n)	Triangle read by rows: T(n,k) is the number of partitions of n having k perfect square parts (0<=k<=n).
A263234	{1, 0, 1, 1, 0, 1, 0, 2, 0, 1, 2, 0, 2, 0, 1}	A000041(n)	Triangle read by rows: T(n,k) is the number of partitions of n having k triangular number parts (0<=k<=n).
A264391	{1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1}	A000041(n)	Triangle read by rows: T(n,k) is the number of partitions of n having k perfect cube parts (0<=k<=n).
A264394	{1, 0, 1, 1, 0, 1, 0, 2, 0, 1, 2, 0, 2, 0, 1}	A000041(n)	Triangle read by rows: T(n,k) is the number of partitions of n having k Mersenne number parts (0<=k<=n).
A264403	{1, 0, 1, 1, 0, 1, 1, 0, 0, 2, 2, 0, 1, 0, 2}	A000041(n)	Triangle read by rows: T(n,k) is the number of partitions of n in which the sum of the parts of multiplicity 1 is equal to k (0<=k<=n).
A264405	{1, 1, 0, 1, 0, 1, 2, 0, 0, 1, 2, 0, 2, 0, 1}	A000041(n)	Triangle read by rows: T(n,k) is the number of integer partitions of n having k repeated parts (each occurrence is counted).
A276422	{1, 0, 1, 2, 0, 0, 1, 1, 0, 1, 4, 0, 0, 0, 1}	A000041(n)	Triangle read by rows: T(n,k) is the number of partitions of n for which the sum of its odd singletons is k (0<=k<=n). A singleton in a partition is a part that occurs exactly once.
A276424	{1, 1, 0, 1, 0, 1, 2, 0, 1, 0, 3, 0, 1, 0, 1}	A000041(n)	Triangle read by rows: T(n,k) is the number of partitions of n for which the sum of its even singletons is k (0<=k<=n). A singleton in a partition is a part that occurs exactly once.
A307431	{1, 0, 1, 0, 1, 1, 0, 1, 0, 2, 0, 1, 1, 2, 1}	A000041(n)	Number T(n,k) of partitions of n into parts whose bitwise OR equals k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
A307432	{1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1}	A000041(n)	Number T(n,k) of partitions of n into parts whose bitwise AND equals k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
A308151	{1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1}	A000041(n)	Triangular array: each row partitions the partitions of n into n parts; of which the k-th part is the number of partitions having stay number k-1; see Comments.
A322393	{1, 0, 1, 1, 1, 0, 2, 1, 0, 0, 3, 1, 1, 0, 0}	A000041(n)	Regular triangle read by rows where T(n,k) is the number of integer partitions of n with edge-connectivity k, for 0 <= k <= n.
A325165	{1, 0, 1, 0, 0, 2, 0, 0, 0, 3, 0, 1, 0, 0, 4}	A000041(n)	Regular triangle read by rows where T(n,k) is the number of integer partitions of n whose inner lining partition has last (smallest) part equal to k.
A325188	{1, 0, 1, 0, 2, 0, 0, 2, 1, 0, 0, 2, 3, 0, 0}	A000041(n)	Regular triangle read by rows where T(n,k) is the number of integer partitions of n with origin-to-boundary graph-distance equal to k.
A325189	{1, 0, 1, 0, 0, 2, 0, 0, 1, 2, 0, 0, 0, 3, 2}	A000041(n)	Regular triangle read by rows where T(n,k) is the number of integer partitions of n with maximum origin-to-boundary graph-distance equal to k.
A325192	{1, 1, 0, 0, 2, 0, 0, 1, 2, 0, 1, 0, 2, 2, 0}	A000041(n)	Regular triangle read by rows where T(n,k) is the number of integer partitions of n such that the difference between the length of the minimal square containing and the maximal square contained in the Young diagram is k.
A325200	{1, 1, 0, 0, 2, 0, 1, 0, 2, 0, 0, 3, 0, 2, 0}	A000041(n)	Regular triangle read by rows where T(n,k) is the number of integer partitions of n such that the difference between the length of the minimal triangular partition containing and the maximal triangular partition contained in the Young diagram is k.
A325268	{1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 3, 0, 1}	A000041(n)	Triangle read by rows where T(n,k) is the number of integer partitions of n with omicron k.
A325280	{1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1}	A000041(n)	Triangle read by rows where T(n,k) is the number of integer partitions of n with adjusted frequency depth k.
A325458	{1, 0, 1, 0, 0, 2, 0, 0, 0, 3, 0, 0, 0, 1, 4}	A000041(n)	Triangle read by rows where T(n,k) is the number of integer partitions of n with largest hook of size k, i.e., with (largest part) + (number of parts) - 1 = k.
A325466	{1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 3, 1, 0, 0}	A000041(n)	Triangle read by rows where T(n,k) is the number of reversed integer partitions of n with k distinct differences of any degree > 0.
A339737	{1, 1, 0, 1, 1, 0, 2, 0, 1, 0, 2, 1, 1, 1, 0}	A000041(n)	Triangle read by rows where T(n,k) is the number of integer partitions of n with greatest gap k.
A344612	{1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1}	A000041(n)	Triangle read by rows where T(n,k) is the number of integer partitions of n with reverse-alternating sum k ranging from -n to n in steps of 2.
A353315	{1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1}	A000041(n)	Triangle read by rows where T(n,k) is the number of integer partitions of n with k parts on or below the diagonal (weak non-excedances).
A353836	{1, 0, 1, 0, 2, 0, 0, 2, 1, 0, 0, 4, 1, 0, 0}	A000041(n)	Triangle read by rows where T(n,k) is the number of integer partitions of n with k distinct run-sums.
A353846	{1, 0, 1, 0, 1, 1, 0, 2, 1, 0, 0, 2, 2, 1, 0}	A000041(n)	Triangle read by rows where T(n,k) is the number of integer partitions of n with partition run-sum trajectory of length k.
A357637	{1, 0, 1, 0, 0, 2, 0, 0, 1, 2, 0, 0, 1, 1, 3}	A000041(n)	Triangle read by rows where T(n,k) is the number of integer partitions of n with half-alternating sum k, where k ranges from -n to n in steps of 2.
A357638	{1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 3, 1, 1}	A000041(n)	Triangle read by rows where T(n,k) is the number of integer partitions of n with skew-alternating sum k, where k ranges from -n to n in steps of 2.
A357704	{1, 0, 1, 0, 0, 2, 0, 0, 1, 2, 0, 0, 2, 0, 3}	A000041(n)	Triangle read by rows where T(n,k) is the number of reversed integer partitions of n with half-alternating sum k, where k ranges from -n to n in steps of 2.
A357705	{1, 0, 1, 0, 1, 1, 0, 2, 0, 1, 0, 2, 2, 0, 1}	A000041(n)	Triangle read by rows where T(n,k) is the number of reversed integer partitions of n with skew-alternating sum k, where k ranges from -n to n in steps of 2.
A360672	{1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 3, 1, 0}	A000041(n)	Triangle read by rows where T(n,k) is the number of integer partitions of n whose left half (exclusive) sums to k, where k ranges from 0 to n.
A360675	{1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 2, 2, 0, 0}	A000041(n)	Triangle read by rows where T(n,k) is the number of integer partitions of n whose right half (exclusive) sums to k, where k ranges from 0 to n.
A363946	{1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 3, 0, 1}	A000041(n)	Triangle read by rows where T(n,k) is the number of integer partitions of n with high mean k.
A363952	{1, 0, 1, 0, 1, 1, 0, 2, 0, 1, 0, 3, 1, 0, 1}	A000041(n)	Number of integer partitions of n with low mode k.
A363953	{1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 2, 1, 1, 1}	A000041(n)	Number of integer partitions of n with high mode k.
A364060	{1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 2, 2, 0, 1}	A000041(n)	Triangle read by rows where T(n,k) is the number of integer partitions of n with rounded mean k.
A364916	{1, 1, 0, 1, 1, 0, 2, 0, 1, 0, 2, 1, 1, 1, 0}	A000041(n)	Array read by anti-diagonals downwards where A(n,k) is the number of ways to write n as a nonnegative linear combination of the parts of a strict integer partition of k.
A365676	{1, 0, 1, 0, 2, 0, 0, 2, 1, 0, 0, 3, 2, 0, 0}	A000041(n)	Triangle read by rows: T(n, k) is the number of partitions of n having exactly k distinct parts, for 0 <= k <= n.
A365921	{1, 1, 0, 1, 1, 0, 2, 0, 1, 0, 2, 0, 1, 2, 0}	A000041(n)	Triangle read by rows where T(n,k) is the number of integer partitions y of n such that k is the greatest member of {0..n} that is not the sum of any nonempty submultiset of y.
A365923	{1, 1, 0, 1, 1, 0, 2, 0, 1, 0, 2, 1, 1, 1, 0}	A000041(n)	Triangle read by rows where T(n,k) is the number of integer partitions of n with exactly k distinct non-subset-sums.
A367582	{1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1}	A000041(n)	Triangle read by rows where T(n,k) is the number of integer partitions of n whose multiset multiplicity kernel (in which each multiplicity becomes the least element of that multiplicity), sums to k.
A008284	{1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1}	A000041(n+1)	Triangle of partition numbers: T(n,k) = number of partitions of n in which the greatest part is k, 1 <= k <= n. Also number of partitions of n into k positive parts, 1 <= k <= n.
A026794	{1, 1, 1, 2, 0, 1, 3, 1, 0, 1, 5, 1, 0, 0, 1}	A000041(n+1)	Triangular array T read by rows: T(n,k) = number of partitions of n in which least part is k, 1<=k<=n.
A049597	{1, 0, 2, 0, 0, 3, 0, 0, 1, 4, 0, 0, 0, 2, 5}	A000041(n+1)	Triangular array T(n,k) in which k-th column gives coefficients of sum of Gaussian polynomials [k,m] for m=0..k.
A052249	{1, 1, 1, 0, 2, 1, 0, 1, 3, 1, 0, 0, 2, 4, 1}	A000041(n+1)	Triangle T(n,k) (n >= 1, k >= 1) giving dimension of bigrading of Connes-Moscovici noncocommutative algebra.
A058398	{1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1}	A000041(n+1)	Partition triangle A008284 read from right to left.
A091602	{1, 1, 1, 2, 0, 1, 2, 2, 0, 1, 3, 2, 1, 0, 1}	A000041(n+1)	Triangle: T(n,k) is the number of partitions of n such that some part is repeated k times and no part is repeated more than k times.
A096144	{1, 1, 1, 2, 0, 1, 2, 2, 0, 1, 4, 1, 1, 0, 1}	A000041(n+1)	Triangle T(n,k) = number of partitions of n in which the least part occurs exactly k times, k=1..n.
A096771	{1, 0, 2, 0, 1, 2, 0, 1, 2, 2, 0, 0, 3, 2, 2}	A000041(n+1)	Triangle read by rows: T(n,m) is the number of partitions of n that (just) fit inside an m X m box, but not in an (m-1) X (m-1) box. Partitions of n with Max(max part, length) = m.
A097364	{1, 2, 0, 2, 1, 0, 3, 1, 1, 0, 2, 3, 1, 1, 0}	A000041(n+1)	Triangle read by rows, 0 <= k < n: T(n,k) = number of partitions of n such that the differences between greatest and smallest parts are k.
A114087	{1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1}	A000041(n+1)	Triangle read by rows: T(n,k) is the number of partitions of n whose tails below their Durfee squares have size k (n>=1; 0<=k<=n-1).
A114088	{1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1}	A000041(n+1)	Triangle read by rows: T(n,k) is number of partitions of n whose tail below its Durfee square has k parts (n >= 1; 0 <= k <= n-1).
A115723	{1, 0, 2, 0, 1, 2, 0, 0, 2, 3, 0, 0, 1, 4, 2}	A000041(n+1)	Table of partitions of n with maximum rectangle k.
A116861	{1, 1, 1, 1, 0, 2, 1, 1, 1, 2, 1, 0, 2, 1, 3}	A000041(n+1)	Triangle read by rows: T(n,k) is the number of partitions of n such that the sum of the parts, counted without multiplicities, is equal to k (n>=1, k>=1).
A124943	{1, 1, 1, 2, 0, 1, 3, 1, 0, 1, 4, 2, 0, 0, 1}	A000041(n+1)	Table read by rows: number of partitions of n with k as low median.
A124944	{1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1}	A000041(n+1)	Table, number of partitions of n with k as high median.
A130162	{1, 1, 1, 1, 0, 2, 1, 1, 0, 3, 1, 0, 0, 0, 6}	A000041(n+1)	Triangle read by rows: A051731 * A000837 as a diagonalized matrix.
A132825	{1, 0, 2, 0, 0, 3, 0, 0, 0, 5, 0, 0, 0, 0, 7}	A000041(n+1)	Triangle read by rows: zeros except for right border which are the partition numbers A000041.
A133121	{1, 1, 1, 2, 0, 1, 2, 2, 0, 1, 3, 2, 1, 0, 1}	A000041(n+1)	Triangle T(n,k) read by rows = number of partitions of n such that number of parts minus number of distinct parts is equal to k, k = 0..n-1.
A134979	{1, 0, 2, 0, 1, 2, 0, 1, 1, 3, 0, 0, 3, 2, 2}	A000041(n+1)	Triangle read by rows: T(n,k) = number of partitions of n where the maximum number of objects in partitions of any given size is k.
A135486	{1, 0, 2, 1, 0, 2, 1, 1, 0, 3, 5, 0, 0, 0, 2}	A000041(n+1)	Triangle read by rows: T(n,k) = number of partitions of n having k-fold symmetry, cf. A085436.
A168532	{1, 1, 1, 2, 0, 1, 3, 1, 0, 1, 6, 0, 0, 0, 1}	A000041(n+1)	Triangle read by rows, A054525 * A168021.
A174067	{1, 1, 1, 2, 0, 1, 3, 1, 0, 1, 4, 1, 1, 0, 1}	A000041(n+1)	Triangle, row sums = A000041 starting (1, 2, 3, 5, 7, ...); derived from finite differences of p(x) = A(x)*A(x^2) = B(x)*B(x^3) = C(x)*C(x^4) = ...
A175010	{1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3}	A000041(n+1)	Triangle generated from INVERT transforms of variants of A080995.
A194799	{1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1}	A000041(n+1)	Triangle read by rows: T(n,k) = number of partitions of n that are formed by k sections, k >= 1.
A218907	{1, 2, 0, 2, 0, 1, 2, 0, 2, 1, 2, 0, 2, 2, 1}	A000041(n+1)	Triangle, read by rows, of integer partitions of n by kernel size k.
A230025	{1, 0, 2, 1, 0, 2, 1, 2, 0, 2, 1, 2, 2, 0, 2}	A000041(n+1)	Triangular array: t(n, k) = number of occurrences of k as the number of outliers in all the partitions of n.
A237513	{1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1}	A000041(n+1)	T(n,k) = number of maximal horizontal rectangles that contain the Durfee square for partitions of n that consist of k nodes, 1 <= k <= n; triangular array read by rows.
A265247	{1, 2, 0, 2, 0, 1, 3, 0, 1, 1, 2, 0, 2, 2, 1}	A000041(n+1)	Triangle read by rows: T(n,k) is the number of partitions of n in which the 2nd smallest part is k when the partition has at least 2 distinct parts and 0 otherwise; (n>=1, 0 <= k <= n).
A268189	{1, 2, 0, 2, 0, 1, 3, 0, 1, 1, 2, 0, 1, 2, 2}	A000041(n+1)	Triangle read by rows: T(n,k) is the number of partitions of n for which the sum of the parts larger than the smallest part is k (n>=1, 0<=k<=n-1).
A279044	{1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1}	A000041(n+1)	Irregular triangular array: T(n,i) = number of partitions of n having crossover part k; see Comments.
A303810	{1, 1, 1, 1, 0, 2, 1, 0, 1, 3, 1, 0, 0, 1, 5}	A000041(n+1)	Mirror image of the triangle A026794.
A329746	{1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 3, 0}	A000041(n+1)	Triangle read by rows where T(n,k) is the number of integer partitions of n > 0 with runs-resistance k, 0 <= k <= n - 1.
A330374	{1, 0, 2, 1, 0, 2, 1, 2, 0, 2, 1, 2, 2, 0, 2}	A000041(n+1)	Triangle read by rows: T(n,k) is the number of partitions of n whose absolute value of Dyson's rank is equal to k, with 0 <= k < n.
A341418	{1, 1, 1, 0, 2, 1, 0, 1, 3, 1, -1, 0, 3, 4, 1}	A000041(n+1)	Triangle read by rows: T(n, m) gives the sum of the weights of weighted compositions of n with m parts from generalized pentagonal numbers {A001318(k)}_{k>=1}.
A365658	{1, 1, 1, 1, 0, 2, 1, 1, 1, 2, 1, 0, 2, 0, 4}	A000041(n+1)	Triangle read by rows where T(n,k) is the number of integer partitions of n with k distinct possible sums of nonempty submultisets.
A366745	{1, 0, 2, 1, 0, 2, 1, 2, 0, 2, 1, 2, 2, 0, 2}	A000041(n+1)	Triangular array, read by rows: T(n,k) = number of partitions p of n such that d(p, p') = 2k, where p' = contraconjugate of p, and d is the distance function defined in A366156.