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# Symmetric Functions in Non-Commuting Variables

AUTHORS:

- Travis Scrimshaw (08-04-2013): Initial version

class `sage.combinat.ncsym.ncsym.``SymmetricFunctionsNonCommutingVariables`(R)

  Bases: `sage.structure.unique_representation.UniqueRepresentation`, `sage.structure.parent.Parent`

  Symmetric functions in non-commutative variables.

  The ring of symmetric functions in non-commutative variables, which is not to be confused with the `non-commutative symmetric functions`, is the ring of all bounded-degree noncommutative power series in countably many indeterminates (i.e., elements in \(R \langle \langle x_1, x_2, x_3, \ldots \rangle \rangle\) of bounded degree) which are invariant with respect to the action of the symmetric group \(S_{\infty}\) on the indices of the indeterminates. It can be regarded as a direct limit over all \(n \to \infty\) of rings of \(S_n\)-invariant polynomials in \(n\) non-commuting variables (that is, \(S_n\)-invariant elements of \(R\langle x_1, x_2, \ldots, x_n \rangle\)).

  This ring is implemented as a Hopf algebra whose basis elements are indexed by set parititions.

  Let \(A = \{A_1, A_2, \ldots, A_r\}\) be a set partition of the integers \(\{ 1, 2, \ldots, k \}\). A monomial basis element indexed by \(A\) represents the sum of monomials \(x_{i_1} x_{i_2} \cdots x_{i_k}\) where \(i_c = i_d\) if and only if \(c\) and \(d\) are in the same part \(A_i\) for some \(i\).

  The \(k\)-th graded component of the ring of symmetric functions in non-commutative variables has its dimension equal to the number of set partitions of \(k\). (If we work, instead, with finitely many – say, \(n\) – variables, then its dimension is equal to the number of set partitions of \(k\) where the number of parts is at most \(n\).)

  Note

  All set partitions are considered standard, a set partition of \([n]\) for some \(n\), unless otherwise stated.

  REFERENCES:

  [BZ05] | N. Bergeron, M. Zabrocki. The Hopf algebra of symmetric functions and quasisymmetric functions in non-commutative variables are free and cofree. (2005). Arxiv math/0509265v3.

  [BHRZ06] | N. Bergeron, C. Hohlweg, M. Rosas, M. Zabrocki. Grothendieck bialgebras, partition lattices, and symmetric functions in noncommutative variables. Electronic Journal of Combinatorics. 13 (2006).

  [RS06] | M. Rosas, B. Sagan. Symmetric functions in noncommuting variables. Trans. Amer. Math. Soc. 358 (2006). no. 1, 215-232. Arxiv math/0208168.

  [BRRZ08] | N. Bergeron, C. Reutenauer, M. Rosas, M. Zabrocki. Invariants and coinvariants of the symmetric group in noncommuting variables. Canad. J. Math. 60 (2008). 266-296. http://www.lacim.uqam.ca/~christo/Publi%C3%A9s/2008/NBergeronRosasZabrocki.pdf

  EXAMPLES:

  We begin by first creating the ring of \(NCSym\) and the bases that are analogues of the usual symmetric functions:

      sage: NCSym = SymmetricFunctionsNonCommutingVariables(QQ)
      sage: m = NCSym.m()
      sage: e = NCSym.e()
      sage: h = NCSym.h()
      sage: p = NCSym.p()
      sage: m
      Symmetric functions in non-commuting variables over the Rational Field in the monomial basis

  The basis is indexed by set partitions, so we create a few elements and convert them between these bases:

      sage: elt = m(SetPartition([[1,3],[2]])) - 2*m(SetPartition([[1],[2]])); elt
      -2*m{{1}, {2}} + m{{1, 3}, {2}}
      sage: e(elt)
      1/2*e{{1}, {2, 3}} - 2*e{{1, 2}} + 1/2*e{{1, 2}, {3}} - 1/2*e{{1, 2, 3}} - 1/2*e{{1, 3}, {2}}
      sage: h(elt)
      -4*h{{1}, {2}} - 2*h{{1}, {2}, {3}} + 1/2*h{{1}, {2, 3}} + 2*h{{1, 2}}
       + 1/2*h{{1, 2}, {3}} - 1/2*h{{1, 2, 3}} + 3/2*h{{1, 3}, {2}}
      sage: p(elt)
      -2*p{{1}, {2}} + 2*p{{1, 2}} - p{{1, 2, 3}} + p{{1, 3}, {2}}
      sage: m(p(elt))
      -2*m{{1}, {2}} + m{{1, 3}, {2}}

      sage: elt = p(SetPartition([[1,3],[2]])) - 4*p(SetPartition([[1],[2]])) + 2; elt
      2*p{} - 4*p{{1}, {2}} + p{{1, 3}, {2}}
      sage: e(elt)
      2*e{} - 4*e{{1}, {2}} + e{{1}, {2}, {3}} - e{{1, 3}, {2}}
      sage: m(elt)
      2*m{} - 4*m{{1}, {2}} - 4*m{{1, 2}} + m{{1, 2, 3}} + m{{1, 3}, {2}}
      sage: h(elt)
      2*h{} - 4*h{{1}, {2}} - h{{1}, {2}, {3}} + h{{1, 3}, {2}}
      sage: p(m(elt))
      2*p{} - 4*p{{1}, {2}} + p{{1, 3}, {2}}

  There is also a shorthand for creating elements. We note that we must use `p[[]]` to create the empty set partition due to python’s syntax.

      sage: eltm = m[[1,3],[2]] - 3*m[[1],[2]]; eltm
      -3*m{{1}, {2}} + m{{1, 3}, {2}}
      sage: elte = e[[1,3],[2]]; elte
      e{{1, 3}, {2}}
      sage: elth = h[[1,3],[2,4]]; elth
      h{{1, 3}, {2, 4}}
      sage: eltp = p[[1,3],[2,4]] + 2*p[[1]] - 4*p[[]]; eltp
      -4*p{} + 2*p{{1}} + p{{1, 3}, {2, 4}}

  There is also a natural projection to the usual symmetric functions by letting the variables commute. This projection map preserves the product and coproduct structure. We check that Theorem 2.1 of [RS06] holds:

      sage: Sym = SymmetricFunctions(QQ)
      sage: Sm = Sym.m()
      sage: Se = Sym.e()
      sage: Sh = Sym.h()
      sage: Sp = Sym.p()
      sage: eltm.to_symmetric_function()
      -6*m[1, 1] + m[2, 1]
      sage: Sm(p(eltm).to_symmetric_function())
      -6*m[1, 1] + m[2, 1]
      sage: elte.to_symmetric_function()
      2*e[2, 1]
      sage: Se(h(elte).to_symmetric_function())
      2*e[2, 1]
      sage: elth.to_symmetric_function()
      4*h[2, 2]
      sage: Sh(m(elth).to_symmetric_function())
      4*h[2, 2]
      sage: eltp.to_symmetric_function()
      -4*p[] + 2*p[1] + p[2, 2]
      sage: Sp(e(eltp).to_symmetric_function())
      -4*p[] + 2*p[1] + p[2, 2]

  `a_realization`()

    Return the realization of the powersum basis of `self`.

    OUTPUT:

    - The powersum basis of symmetric functions in non-commuting variables.

    EXAMPLES:

        sage: SymmetricFunctionsNonCommutingVariables(QQ).a_realization()
        Symmetric functions in non-commuting variables over the Rational Field in the powersum basis

  `dual`()

    Return the dual Hopf algebra of the symmetric functions in non-commuting variables.

    EXAMPLES:

        sage: SymmetricFunctionsNonCommutingVariables(QQ).dual()
        Dual symmetric functions in non-commuting variables over the Rational Field

  class `elementary`(NCSym)

    Bases: `sage.combinat.ncsym.bases.NCSymBasis_abstract`

    The Hopf algebra of symmetric functions in non-commuting variables in the elementary basis.

    EXAMPLES:

        sage: NCSym = SymmetricFunctionsNonCommutingVariables(QQ)
        sage: e = NCSym.e()

    class `Element`(M, x)

      Bases: `sage.combinat.free_module.CombinatorialFreeModuleElement`

      An element in the elementary basis of \(NCSym\).

      `omega`()

        Return the involution \(\omega\) applied to `self`.

        The involution \(\omega\) on \(NCSym\) is defined by \(\omega(\mathbf{e}_A) = \mathbf{h}_A\).

        OUTPUT:

        - an element in the basis `self`

        EXAMPLES:

            sage: NCSym = SymmetricFunctionsNonCommutingVariables(QQ)
            sage: e = NCSym.e()
            sage: h = NCSym.h()
            sage: elt = e[[1,3],[2]].omega(); elt
            2*e{{1}, {2}, {3}} - e{{1, 3}, {2}}
            sage: elt.omega()
            e{{1, 3}, {2}}
            sage: h(elt)
            h{{1, 3}, {2}}

      `to_symmetric_function`()

        The projection of `self` to the symmetric functions.

        Take a symmetric function in non-commuting variables expressed in the \(\mathbf{e}\) basis, and return the projection of expressed in the elementary basis of symmetric functions.

        The map \(\chi \colon NCSym \to Sym\) is given by

        \[\mathbf{e}_A \mapsto e_{\lambda(A)} \prod_i \lambda(A)_i!\]

        where \(\lambda(A)\) is the partition associated with \(A\) by taking the sizes of the parts.

        OUTPUT:

        - An element of the symmetric functions in the elementary basis

        EXAMPLES:

            sage: e = SymmetricFunctionsNonCommutingVariables(QQ).e()
            sage: e[[1,3],[2]].to_symmetric_function()
            2*e[2, 1]
            sage: e[[1],[3],[2]].to_symmetric_function()
            e[1, 1, 1]

  class `SymmetricFunctionsNonCommutingVariables.``homogeneous`(NCSym)

    Bases: `sage.combinat.ncsym.bases.NCSymBasis_abstract`

    The Hopf algebra of symmetric functions in non-commuting variables in the homogeneous basis.

    EXAMPLES:

        sage: NCSym = SymmetricFunctionsNonCommutingVariables(QQ)
        sage: h = NCSym.h()

    class `Element`(M, x)

      Bases: `sage.combinat.free_module.CombinatorialFreeModuleElement`

      An element in the homogeneous basis of \(NCSym\).

      `omega`()

        Return the involution \(\omega\) applied to `self`.

        The involution \(\omega\) on \(NCSym\) is defined by \(\omega(\mathbf{h}_A) = \mathbf{e}_A\).

        OUTPUT:

        - an element in the basis `self`

        EXAMPLES:

            sage: NCSym = SymmetricFunctionsNonCommutingVariables(QQ)
            sage: h = NCSym.h()
            sage: e = NCSym.e()
            sage: elt = h[[1,3],[2]].omega(); elt
            2*h{{1}, {2}, {3}} - h{{1, 3}, {2}}
            sage: elt.omega()
            h{{1, 3}, {2}}
            sage: e(elt)
            e{{1, 3}, {2}}

      `to_symmetric_function`()

        The projection of `self` to the symmetric functions.

        Take a symmetric function in non-commuting variables expressed in the \(\mathbf{h}\) basis, and return the projection of expressed in the complete basis of symmetric functions.

        The map \(\chi \colon NCSym \to Sym\) is given by

        \[\mathbf{h}_A \mapsto h_{\lambda(A)} \prod_i \lambda(A)_i!\]

        where \(\lambda(A)\) is the partition associated with \(A\) by taking the sizes of the parts.

        OUTPUT:

        - An element of the symmetric functions in the complete basis

        EXAMPLES:

            sage: h = SymmetricFunctionsNonCommutingVariables(QQ).h()
            sage: h[[1,3],[2]].to_symmetric_function()
            2*h[2, 1]
            sage: h[[1],[3],[2]].to_symmetric_function()
            h[1, 1, 1]

  class `SymmetricFunctionsNonCommutingVariables.``monomial`(NCSym)

    Bases: `sage.combinat.ncsym.bases.NCSymBasis_abstract`

    The Hopf algebra of symmetric functions in non-commuting variables in the monomial basis.

    EXAMPLES:

        sage: NCSym = SymmetricFunctionsNonCommutingVariables(QQ)
        sage: m = NCSym.m()

    class `Element`(M, x)

      Bases: `sage.combinat.free_module.CombinatorialFreeModuleElement`

      An element in the monomial basis of \(NCSym\).

      `expand`(n, alphabet='x')

        Expand `self` written in the monomial basis in \(n\) non-commuting variables.

        INPUT:

        - `n` – an integer
        - `alphabet` – (default: `'x'`) a string

        OUTPUT:

        - The symmetric function of `self` expressed in the `n` non-commuting variables described by `alphabet`.

        EXAMPLES:

            sage: m = SymmetricFunctionsNonCommutingVariables(QQ).monomial()
            sage: m[[1,3],[2]].expand(4)
            x0*x1*x0 + x0*x2*x0 + x0*x3*x0 + x1*x0*x1 + x1*x2*x1 + x1*x3*x1
             + x2*x0*x2 + x2*x1*x2 + x2*x3*x2 + x3*x0*x3 + x3*x1*x3 + x3*x2*x3

        One can use a different set of variables by using the optional argument `alphabet`:

            sage: m[[1],[2,3]].expand(3,alphabet='y')
            y0*y1^2 + y0*y2^2 + y1*y0^2 + y1*y2^2 + y2*y0^2 + y2*y1^2

      `to_symmetric_function`()

        The projection of `self` to the symmetric functions.

        Take a symmetric function in non-commuting variables expressed in the \(\mathbf{m}\) basis, and return the projection of expressed in the monomial basis of symmetric functions.

        The map \(\chi \colon NCSym \to Sym\) is defined by

        \[\mathbf{m}_A \mapsto m_{\lambda(A)} \prod_i n_i(\lambda(A))!\]

        where \(\lambda(A)\) is the partition associated with \(A\) by taking the sizes of the parts and \(n_i(\mu)\) is the multiplicity of \(i\) in \(\mu\).

        OUTPUT:

        - an element of the symmetric functions in the monomial basis

        EXAMPLES:

            sage: m = SymmetricFunctionsNonCommutingVariables(QQ).monomial()
            sage: m[[1,3],[2]].to_symmetric_function()
            m[2, 1]
            sage: m[[1],[3],[2]].to_symmetric_function()
            6*m[1, 1, 1]

    `SymmetricFunctionsNonCommutingVariables.monomial.``coproduct_on_basis`(A)

      Return the coproduct of a monomial basis element.

      INPUT:

      - `A` – a set partition

      OUTPUT:

      - The coproduct applied to the monomial symmetric function in non-commuting variables indexed by `A` expressed in the monomial basis.

      EXAMPLES:

          sage: m = SymmetricFunctionsNonCommutingVariables(QQ).monomial()
          sage: m[[1, 3], [2]].coproduct()
          m{} # m{{1, 3}, {2}} + m{{1}} # m{{1, 2}} + m{{1, 2}} # m{{1}} + m{{1, 3}, {2}} # m{}
          sage: m.coproduct_on_basis(SetPartition([]))
          m{} # m{}
          sage: m.coproduct_on_basis(SetPartition([[1,2,3]]))
          m{} # m{{1, 2, 3}} + m{{1, 2, 3}} # m{}
          sage: m[[1,5],[2,4],[3,7],[6]].coproduct()
          m{} # m{{1, 5}, {2, 4}, {3, 7}, {6}} + m{{1}} # m{{1, 5}, {2, 4}, {3, 6}}
           + 2*m{{1, 2}} # m{{1, 3}, {2, 5}, {4}} + m{{1, 2}} # m{{1, 4}, {2, 3}, {5}}
           + 2*m{{1, 2}, {3}} # m{{1, 3}, {2, 4}} + m{{1, 3}, {2}} # m{{1, 4}, {2, 3}}
           + 2*m{{1, 3}, {2, 4}} # m{{1, 2}, {3}} + 2*m{{1, 3}, {2, 5}, {4}} # m{{1, 2}}
           + m{{1, 4}, {2, 3}} # m{{1, 3}, {2}} + m{{1, 4}, {2, 3}, {5}} # m{{1, 2}}
           + m{{1, 5}, {2, 4}, {3, 6}} # m{{1}} + m{{1, 5}, {2, 4}, {3, 7}, {6}} # m{}

    `SymmetricFunctionsNonCommutingVariables.monomial.``dual_basis`()

      Return the dual basis to the monomial basis.

      OUTPUT:

      - the \(\mathbf{w}\) basis of the dual Hopf algebra

      EXAMPLES:

          sage: m = SymmetricFunctionsNonCommutingVariables(QQ).m()
          sage: m.dual_basis()
          Dual symmetric functions in non-commuting variables over the Rational Field in the w basis

    `SymmetricFunctionsNonCommutingVariables.monomial.``duality_pairing`(x, y)

      Compute the pairing between an element of `self` and an element of the dual.

      INPUT:

      - `x` – an element of symmetric functions in non-commuting variables
      - `y` – an element of the dual of symmetric functions in non-commuting variables

      OUTPUT:

      - an element of the base ring of `self`

      EXAMPLES:

          sage: NCSym = SymmetricFunctionsNonCommutingVariables(QQ)
          sage: m = NCSym.m()
          sage: w = m.dual_basis()
          sage: matrix([[m(A).duality_pairing(w(B)) for A in SetPartitions(3)] for B in SetPartitions(3)])
          [1 0 0 0 0]
          [0 1 0 0 0]
          [0 0 1 0 0]
          [0 0 0 1 0]
          [0 0 0 0 1]
          sage: (m[[1,2],[3]] + 3*m[[1,3],[2]]).duality_pairing(2*w[[1,3],[2]] + w[[1,2,3]] + 2*w[[1,2],[3]])
          8

    `SymmetricFunctionsNonCommutingVariables.monomial.``from_symmetric_function`(f)

      Return the image of the symmetric function `f` in `self`.

      This is performed by converting to the monomial basis and extending the method `sum_of_partitions()` linearly. This is a linear map from the symmetric functions to the symmetric functions in non-commuting variables that does not preserve the product or coproduct structure of the Hopf algebra.

      See also

      `to_symmetric_function()`

      INPUT:

      - `f` – an element of the symmetric functions

      OUTPUT:

      - An element of the \(\mathbf{m}\) basis

      EXAMPLES:

          sage: m = SymmetricFunctionsNonCommutingVariables(QQ).m()
          sage: mon = SymmetricFunctions(QQ).m()
          sage: elt = m.from_symmetric_function(mon[2,1,1]); elt
          1/12*m{{1}, {2}, {3, 4}} + 1/12*m{{1}, {2, 3}, {4}} + 1/12*m{{1}, {2, 4}, {3}}
           + 1/12*m{{1, 2}, {3}, {4}} + 1/12*m{{1, 3}, {2}, {4}} + 1/12*m{{1, 4}, {2}, {3}}
          sage: elt.to_symmetric_function()
          m[2, 1, 1]
          sage: e = SymmetricFunctionsNonCommutingVariables(QQ).e()
          sage: elm = SymmetricFunctions(QQ).e()
          sage: e(m.from_symmetric_function(elm[4]))
          1/24*e{{1, 2, 3, 4}}
          sage: h = SymmetricFunctionsNonCommutingVariables(QQ).h()
          sage: hom = SymmetricFunctions(QQ).h()
          sage: h(m.from_symmetric_function(hom[4]))
          1/24*h{{1, 2, 3, 4}}
          sage: p = SymmetricFunctionsNonCommutingVariables(QQ).p()
          sage: pow = SymmetricFunctions(QQ).p()
          sage: p(m.from_symmetric_function(pow[4]))
          p{{1, 2, 3, 4}}
          sage: p(m.from_symmetric_function(pow[2,1]))
          1/3*p{{1}, {2, 3}} + 1/3*p{{1, 2}, {3}} + 1/3*p{{1, 3}, {2}}
          sage: p([[1,2]])*p([[1]])
          p{{1, 2}, {3}}

      Check that \(\chi \circ \widetilde{\chi}\) is the identity on \(Sym\):

          sage: all(m.from_symmetric_function(pow(la)).to_symmetric_function() == pow(la)
          ....:     for la in Partitions(4))
          True

    `SymmetricFunctionsNonCommutingVariables.monomial.``internal_coproduct_on_basis`(A)

      Return the internal coproduct of a monomial basis element.

      The internal coproduct is defined by

      \[\Delta^{\odot}(\mathbf{m}_A) = \sum_{B \wedge C = A} \mathbf{m}_B \otimes \mathbf{m}_C\]

      where we sum over all pairs of set partitions \(B\) and \(C\) whose infimum is \(A\).

      INPUT:

      - `A` – a set partition

      OUTPUT:

      - an element of the tensor square of the \(\mathbf{m}\) basis

      EXAMPLES:

          sage: m = SymmetricFunctionsNonCommutingVariables(QQ).monomial()
          sage: m.internal_coproduct_on_basis(SetPartition([[1,3],[2]]))
          m{{1, 2, 3}} # m{{1, 3}, {2}} + m{{1, 3}, {2}} # m{{1, 2, 3}} + m{{1, 3}, {2}} # m{{1, 3}, {2}}

    `SymmetricFunctionsNonCommutingVariables.monomial.``product_on_basis`(A, B)

      The product on monomial basis elements.

      The product of the basis elements indexed by two set partitions \(A\) and \(B\) is the sum of the basis elements indexed by set partitions \(C\) such that \(C \wedge ([n] | [k]) = A | B\) where \(n = |A|\) and \(k = |B|\). Here \(A \wedge B\) is the infimum of \(A\) and \(B\) and \(A | B\) is the `pipe()` operation. Equivalently we can describe all \(C\) as matchings between the partitions of \(A\) and \(B\) where if \(a \in A\) is matched with \(b \in B\), we take \(a \cup b\) instead of \(a\) and \(b\) in \(C\).

      INPUT:

      - `A`, `B` – set partitions

      OUTPUT:

      - an element of the \(\mathbf{m}\) basis

      EXAMPLES:

          sage: m = SymmetricFunctionsNonCommutingVariables(QQ).monomial()
          sage: A = SetPartition([[1], [2,3]])
          sage: B = SetPartition([[1], [3], [2,4]])
          sage: m.product_on_basis(A, B)
          m{{1}, {2, 3}, {4}, {5, 7}, {6}} + m{{1}, {2, 3, 4}, {5, 7}, {6}}
           + m{{1}, {2, 3, 5, 7}, {4}, {6}} + m{{1}, {2, 3, 6}, {4}, {5, 7}}
           + m{{1, 4}, {2, 3}, {5, 7}, {6}} + m{{1, 4}, {2, 3, 5, 7}, {6}}
           + m{{1, 4}, {2, 3, 6}, {5, 7}} + m{{1, 5, 7}, {2, 3}, {4}, {6}}
           + m{{1, 5, 7}, {2, 3, 4}, {6}} + m{{1, 5, 7}, {2, 3, 6}, {4}}
           + m{{1, 6}, {2, 3}, {4}, {5, 7}} + m{{1, 6}, {2, 3, 4}, {5, 7}}
           + m{{1, 6}, {2, 3, 5, 7}, {4}}
          sage: B = SetPartition([[1], [2]])
          sage: m.product_on_basis(A, B)
          m{{1}, {2, 3}, {4}, {5}} + m{{1}, {2, 3, 4}, {5}}
           + m{{1}, {2, 3, 5}, {4}} + m{{1, 4}, {2, 3}, {5}} + m{{1, 4}, {2, 3, 5}}
           + m{{1, 5}, {2, 3}, {4}} + m{{1, 5}, {2, 3, 4}}
          sage: m.product_on_basis(A, SetPartition([]))
          m{{1}, {2, 3}}

      TESTS:

      We check that we get all of the correct set partitions:

          sage: m = SymmetricFunctionsNonCommutingVariables(QQ).monomial()
          sage: A = SetPartition([[1], [2,3]])
          sage: B = SetPartition([[1], [2]])
          sage: S = SetPartition([[1,2,3], [4,5]])
          sage: AB = SetPartition([[1], [2,3], [4], [5]])
          sage: L = sorted(filter(lambda x: S.inf(x) == AB, SetPartitions(5)), key=str)
          sage: map(list, L) == map(list, sorted(m.product_on_basis(A, B).support(), key=str))
          True

    `SymmetricFunctionsNonCommutingVariables.monomial.``sum_of_partitions`(la)

      Return the sum over all set partitions whose shape is `la` with a fixed coefficient \(C\) defined below.

      Fix a partition \(\lambda\), we define \(\lambda! := \prod_i \lambda_i!\) and \(\lambda^! := \prod_i m_i!\). Recall that \(|\lambda| = \sum_i \lambda_i\) and \(m_i\) is the number of parts of length \(i\) of \(\lambda\). Thus we defined the coefficient as

      \[C := \frac{\lambda! \lambda^!}{|\lambda|!}.\]

      Hence we can define a lift \(\widetilde{\chi}\) from \(Sym\) to \(NCSym\) by

      \[m_{\lambda} \mapsto C \sum_A \mathbf{m}_A\]

      where the sum is over all set partitions whose shape is \(\lambda\).

      INPUT:

      - `la` – an integer partition

      OUTPUT:

      - an element of the \(\mathbf{m}\) basis

      EXAMPLES:

          sage: m = SymmetricFunctionsNonCommutingVariables(QQ).m()
          sage: m.sum_of_partitions(Partition([2,1,1]))
          1/12*m{{1}, {2}, {3, 4}} + 1/12*m{{1}, {2, 3}, {4}} + 1/12*m{{1}, {2, 4}, {3}}
           + 1/12*m{{1, 2}, {3}, {4}} + 1/12*m{{1, 3}, {2}, {4}} + 1/12*m{{1, 4}, {2}, {3}}

      TESTS:

      Check that \(\chi \circ \widetilde{\chi}\) is the identity on \(Sym\):

          sage: m = SymmetricFunctionsNonCommutingVariables(QQ).m()
          sage: mon = SymmetricFunctions(QQ).monomial()
          sage: all(m.from_symmetric_function(mon[la]).to_symmetric_function() == mon[la]
          ....:     for i in range(6) for la in Partitions(i))
          True

  class `SymmetricFunctionsNonCommutingVariables.``powersum`(NCSym)

    Bases: `sage.combinat.ncsym.bases.NCSymBasis_abstract`

    The Hopf algebra of symmetric functions in non-commuting variables in the powersum basis.

    EXAMPLES:

        sage: NCSym = SymmetricFunctionsNonCommutingVariables(QQ)
        sage: p = NCSym.p()

    class `Element`(M, x)

      Bases: `sage.combinat.free_module.CombinatorialFreeModuleElement`

      An element in the powersum basis of \(NCSym\).

      `to_symmetric_function`()

        The projection of `self` to the symmetric functions.

        Take a symmetric function in non-commuting variables expressed in the \(\mathbf{p}\) basis, and return the projection of expressed in the powersum basis of symmetric functions.

        The map \(\chi \colon NCSym \to Sym\) is given by

        \[\mathbf{p}_A \mapsto p_{\lambda(A)}\]

        where \(\lambda(A)\) is the partition associated with \(A\) by taking the sizes of the parts.

        OUTPUT:

        - an element of symmetric functions in the power sum basis

        EXAMPLES:

            sage: p = SymmetricFunctionsNonCommutingVariables(QQ).p()
            sage: p[[1,3],[2]].to_symmetric_function()
            p[2, 1]
            sage: p[[1],[3],[2]].to_symmetric_function()
            p[1, 1, 1]

    `SymmetricFunctionsNonCommutingVariables.powersum.``antipode_on_basis`(A)

      Return the result of the antipode applied to a powersum basis element.

      Let \(A\) be a set partition. The antipode given in [LM2011] is

      \[S(\mathbf{p}_A) = \sum_{\gamma} (-1)^{\ell(\gamma)} \mathbf{p}_{\gamma[A]}\]

      where we sum over all ordered set partitions (i.e. set compositions) of \([\ell(A)]\) and

      \[\gamma[A] = A_{\gamma_1}^{\downarrow} | \cdots | A_{\gamma_{\ell(A)}}^{\downarrow}\]

      is the action of \(\gamma\) on \(A\) defined in `SetPartition.ordered_set_partition_action()`.

      INPUT:

      - `A` – a set partition

      OUTPUT:

      - an element in the basis `self`

      EXAMPLES:

          sage: p = SymmetricFunctionsNonCommutingVariables(QQ).powersum()
          sage: p.antipode_on_basis(SetPartition([[1], [2,3]]))
          p{{1, 2}, {3}}
          sage: p.antipode_on_basis(SetPartition([]))
          p{}
          sage: F = p[[1,3],[5],[2,4]].coproduct()
          sage: F.apply_multilinear_morphism(lambda x,y: x.antipode()*y)
          0

    `SymmetricFunctionsNonCommutingVariables.powersum.``coproduct_on_basis`(A)

      Return the coproduct of a monomial basis element.

      INPUT:

      - `A` – a set partition

      OUTPUT:

      - The coproduct applied to the monomial symmetric function in non-commuting variables indexed by `A` expressed in the monomial basis.

      EXAMPLES:

          sage: p = SymmetricFunctionsNonCommutingVariables(QQ).powersum()
          sage: p[[1, 3], [2]].coproduct()
          p{} # p{{1, 3}, {2}} + p{{1}} # p{{1, 2}} + p{{1, 2}} # p{{1}} + p{{1, 3}, {2}} # p{}
          sage: p.coproduct_on_basis(SetPartition([[1]]))
          p{} # p{{1}} + p{{1}} # p{}
          sage: p.coproduct_on_basis(SetPartition([]))
          p{} # p{}

    `SymmetricFunctionsNonCommutingVariables.powersum.``internal_coproduct_on_basis`(A)

      Return the internal coproduct of a powersum basis element.

      The internal coproduct is defined by

      \[\Delta^{\odot}(\mathbf{p}_A) = \mathbf{p}_A \otimes \mathbf{p}_A\]

      INPUT:

      - `A` – a set partition

      OUTPUT:

      - an element of the tensor square of `self`

      EXAMPLES:

          sage: p = SymmetricFunctionsNonCommutingVariables(QQ).powersum()
          sage: p.internal_coproduct_on_basis(SetPartition([[1,3],[2]]))
          p{{1, 3}, {2}} # p{{1, 3}, {2}}

    `SymmetricFunctionsNonCommutingVariables.powersum.``primitive`(A, i=1)

      Return the primitive associated to `A` in `self`.

      Fix some \(i \in S\). Let \(A\) be an atomic set partition of \(S\), then the primitive \(p(A)\) given in [LM2011] is

      \[p(A) = \sum_{\gamma} (-1)^{\ell(\gamma)-1} \mathbf{p}_{\gamma[A]}\]

      where we sum over all ordered set partitions of \([\ell(A)]\) such that \(i \in \gamma_1\) and \(\gamma[A]\) is the action of \(\gamma\) on \(A\) defined in `SetPartition.ordered_set_partition_action()`. If \(A\) is not atomic, then \(p(A) = 0\).

      See also

      `SetPartition.is_atomic()`

      INPUT:

      - `A` – a set partition
      - `i` – (default: 1) index in the base set for `A` specifying which set of primitives this belongs to

      OUTPUT:

      - an element in the basis `self`

      EXAMPLES:

          sage: p = SymmetricFunctionsNonCommutingVariables(QQ).powersum()
          sage: elt = p.primitive(SetPartition([[1,3], [2]])); elt
          -p{{1, 2}, {3}} + p{{1, 3}, {2}}
          sage: elt.coproduct()
          -p{} # p{{1, 2}, {3}} + p{} # p{{1, 3}, {2}} - p{{1, 2}, {3}} # p{} + p{{1, 3}, {2}} # p{}
          sage: p.primitive(SetPartition([[1], [2,3]]))
          0
          sage: p.primitive(SetPartition([]))
          p{}

  class `SymmetricFunctionsNonCommutingVariables.``q`(NCSym)

    Bases: `sage.combinat.ncsym.bases.NCSymBasis_abstract`

    The Hopf algebra of symmetric functions in non-commuting variables in the \(\mathbf{q}\) basis.

    This basis was defined in [BZ05] as

    \[\mathbf{q}_A = \sum_{A \leq_* B} \mathbf{m}_B\]

    where we sum over all strict coarsenings of the set partition \(A\).

    EXAMPLES:

        sage: NCSym = SymmetricFunctionsNonCommutingVariables(QQ)
        sage: q = NCSym.q()

  class `SymmetricFunctionsNonCommutingVariables.``x`(NCSym)

    Bases: `sage.combinat.ncsym.bases.NCSymBasis_abstract`

    The Hopf algebra of symmetric functions in non-commuting variables in the \(\mathbf{x}\) basis.

    This basis is defined in [BHRZ06] by the formula:

    \[\mathbf{x}_A = \sum_{B \leq A} \mu(B, A) \mathbf{p}_B\]

    and has the following properties:

    \[\mathbf{x}_A \mathbf{x}_B = \mathbf{x}_{A|B}, \quad \quad \Delta^{\odot}(\mathbf{x}_C) = \sum_{A \vee B = C} \mathbf{x}_A \otimes \mathbf{x}_B.\]

    EXAMPLES:

        sage: NCSym = SymmetricFunctionsNonCommutingVariables(QQ)
        sage: x = NCSym.x()

`sage.combinat.ncsym.ncsym.``matchings`(A, B)

  Iterate through all matchings of the sets \(A\) and \(B\).

  EXAMPLES:

      sage: from sage.combinat.ncsym.ncsym import matchings
      sage: list(matchings([1, 2, 3], [-1, -2]))
      [[[1], [2], [3], [-1], [-2]],
       [[1], [2], [3, -1], [-2]],
       [[1], [2], [3, -2], [-1]],
       [[1], [2, -1], [3], [-2]],
       [[1], [2, -1], [3, -2]],
       [[1], [2, -2], [3], [-1]],
       [[1], [2, -2], [3, -1]],
       [[1, -1], [2], [3], [-2]],
       [[1, -1], [2], [3, -2]],
       [[1, -1], [2, -2], [3]],
       [[1, -2], [2], [3], [-1]],
       [[1, -2], [2], [3, -1]],
       [[1, -2], [2, -1], [3]]]

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